SherbyNet - /page/wikibook/entry.mu

<!DOCTYPE html>

<title>Well-posed problem</title>

</head>

<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Well-posed problem</span></span>

</h1>

</header>

<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>well-posed problem</b> is one for which the following properties hold:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup>

</p>

<ol><li>The problem has a solution</li>

<li>The solution is <a href="Uniqueness_quantification" title="Uniqueness quantification">unique</a></li>

<li>The solution's behavior changes <a href="Continuous_function" title="Continuous function">continuously</a> with the <a href="Initial_condition" title="Initial condition">initial conditions</a>.</li></ol>

<p>Examples of <a href="Archetypal" class="mw-redirect" title="Archetypal">archetypal</a> well-posed problems include the <a href="Laplace's_equation#Boundary_conditions" title="Laplace's equation">Dirichlet problem for Laplace's equation</a>, and the <a href="Heat_equation" title="Heat equation">heat equation</a> with specified initial conditions. These might be regarded as 'natural' problems in that there are physical processes modelled by these problems.

</p><p>Problems that are not well-posed in the sense above are termed <b>ill-posed</b>. A simple example is a <a href="Global_optimization" title="Global optimization">global optimization</a> problem, because the location of the optima is generally not a continuous function of the parameters specifying the objective, even when the objective itself is a smooth function of those parameters. <a href="Inverse_problem" title="Inverse problem">Inverse problems</a> are often ill-posed; for example, the inverse heat equation, deducing a previous distribution of temperature from final data, is not well-posed in that the solution is highly sensitive to changes in the final data.

</p><p>Continuum models must often be <a href="Discretization" title="Discretization">discretized</a> in order to obtain a numerical solution. While solutions may be continuous with respect to the initial conditions, they may suffer from <a href="Numerical_instability" class="mw-redirect" title="Numerical instability">numerical instability</a> when solved with finite <a href="Accuracy_and_precision" title="Accuracy and precision">precision</a>, or with <a href="Error" title="Error">errors</a> in the data.

</p>

<div class="mw-heading mw-heading2"><h2 id="Conditioning">Conditioning</h2></div>

<p>Even if a problem is well-posed, it may still be <i><a href="Ill-conditioned" class="mw-redirect" title="Ill-conditioned">ill-conditioned</a></i>, meaning that a small error in the initial data can result in much larger errors in the answers. Problems in nonlinear <a href="Complex_systems" class="mw-redirect" title="Complex systems">complex systems</a> (so-called <a href="Chaos_theory" title="Chaos theory">chaotic</a> systems) provide well-known examples of instability. An ill-conditioned problem is indicated by a large <a href="Condition_number" title="Condition number">condition number</a>.

</p><p>If the problem is well-posed, then it stands a good chance of solution on a computer using a <a href="Numerical_stability" title="Numerical stability">stable algorithm</a>. If it is not well-posed, it needs to be re-formulated for numerical treatment. Typically this involves including additional assumptions, such as smoothness of solution. This process is known as <i><a href="Regularization_(mathematics)" title="Regularization (mathematics)">regularization</a></i>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <a href="Tikhonov_regularization" class="mw-redirect" title="Tikhonov regularization">Tikhonov regularization</a> is one of the most commonly used for regularization of linear ill-posed problems.

</p>

<div class="mw-heading mw-heading2"><h2 id="Existence_of_local_solutions">Existence of local solutions</h2></div>

<p>The existence of local solutions is often an important part of the well-posedness problem, and it is the foundation of many estimate methods, for example the energy method below.

</p><p>There are many results on this topic. For example, the <a href="Cauchy%E2%80%93Kowalevski_theorem" class="mw-redirect" title="Cauchy–Kowalevski theorem">Cauchy–Kowalevski theorem</a> for Cauchy initial value problems essentially states that if the terms in a <a href="Partial_differential_equation" title="Partial differential equation">partial differential equation</a> are all made up of <a href="Analytic_function" title="Analytic function">analytic functions</a> and a certain transversality condition is satisfied (the <a href="Hyperplane" title="Hyperplane">hyperplane</a> or more generally <a href="Hypersurface" title="Hypersurface">hypersurface</a> where the initial data are posed must be non-<a href="Method_of_characteristics" title="Method of characteristics">characteristic</a> with respect to the partial differential operator), then on certain regions, there necessarily exist solutions which are as well analytic functions. This is a fundamental result in the study of analytic partial differential equations. Surprisingly, the theorem does not hold in the setting of smooth functions; an <a href="Lewy's_example" title="Lewy's example">example</a> discovered by <a href="Hans_Lewy" title="Hans Lewy">Hans Lewy</a> in 1957 consists of a linear partial differential equation whose coefficients are smooth (i.e., have derivatives of all orders) but not analytic for which no solution exists. So the Cauchy–Kowalevski theorem is necessarily limited in its scope to analytic functions.

</p>

<div class="mw-heading mw-heading2"><h2 id="Energy_method">Energy method</h2></div>

<p>The energy method is useful for establishing both uniqueness and continuity with respect to initial conditions (i.e. it does not establish existence).

The method is based upon deriving an upper bound of an energy-like functional for a given problem.

</p><p><b>Example</b>:

Consider the diffusion equation on the unit interval with homogeneous <a href="Dirichlet_boundary_conditions" class="mw-redirect" title="Dirichlet boundary conditions">Dirichlet boundary conditions</a> and suitable initial data <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

</semantics>

</math></span><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"></span> (e.g. for which <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(0)=f(1)=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(0)=f(1)=0}</annotation>

</semantics>

</math></span><img src="./1fba537b58006d1cba240a36302fe7b640aa1768.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.86ex; height:2.843ex;" alt="{\displaystyle f(0)=f(1)=0}" loading="lazy"></span>).

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}u_{t}&=Du_{xx},&&0<x<1,\,t>0,\,D>0,\\u(x,0)&=f(x),\\u(0,t)&=0,\\u(1,t)&=0,\\\end{aligned}}}">

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}u_{t}&=Du_{xx},&&0<x<1,\,t>0,\,D>0,\\u(x,0)&=f(x),\\u(0,t)&=0,\\u(1,t)&=0,\\\end{aligned}}}</annotation>

</semantics>

</math></span></span>

</p><p>Multiply the equation <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{t}=Du_{xx}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u_{t}=Du_{xx}}</annotation>

</semantics>

</math></span><img src="./59c17a3abf11e815d1d26ca780cf73e0fdc87707.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.621ex; height:2.509ex;" alt="{\displaystyle u_{t}=Du_{xx}}" loading="lazy"></span> by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

</semantics>

</math></span><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"></span> and integrate in space over the unit interval to obtain

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&&\int _{0}^{1}uu_{t}dx&=D\int _{0}^{1}uu_{xx}dx\\\Longrightarrow &&\int _{0}^{1}{\frac {1}{2}}\partial _{t}u^{2}dx&=Duu_{x}{\Big |}_{0}^{1}-D\int _{0}^{1}(u_{x})^{2}dx\\\Longrightarrow &&{\frac {1}{2}}\partial _{t}\|u\|_{2}^{2}&=0-D\int _{0}^{1}(u_{x})^{2}dx\leq 0\end{aligned}}}">

<mtr>

<mtd>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mrow>

</msubsup>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mtd>

<mtd>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&&\int _{0}^{1}uu_{t}dx&=D\int _{0}^{1}uu_{xx}dx\\\Longrightarrow &&\int _{0}^{1}{\frac {1}{2}}\partial _{t}u^{2}dx&=Duu_{x}{\Big |}_{0}^{1}-D\int _{0}^{1}(u_{x})^{2}dx\\\Longrightarrow &&{\frac {1}{2}}\partial _{t}\|u\|_{2}^{2}&=0-D\int _{0}^{1}(u_{x})^{2}dx\leq 0\end{aligned}}}</annotation>

</semantics>

</math></span></span>

</p><p>This tells us that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|u\|_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|u\|_{2}}</annotation>

</semantics>

</math></span><img src="./cb900069dfc0de7529b06d1a5aede3449d0d955b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.709ex; height:2.843ex;" alt="{\displaystyle \|u\|_{2}}" loading="lazy"></span> (<a href="P-norm" class="mw-redirect" title="P-norm">p-norm</a>) cannot grow in time.

By multiplying by two and integrating in time, from <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math></span><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"></span> up to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math></span><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"></span>, one finds

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|u(\cdot ,t)\|_{2}^{2}\leq \|f(\cdot )\|_{2}^{2}}">

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|u(\cdot ,t)\|_{2}^{2}\leq \|f(\cdot )\|_{2}^{2}}</annotation>

</semantics>

</math></span></span>

</p><p>This result is the <b>energy estimate</b> for this problem.

</p><p>To show uniqueness of solutions, assume there are two distinct solutions to the problem, call them <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

</semantics>

</math></span><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math></span><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"></span>, each satisfying the same initial data.

Upon defining <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w=u-v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w=u-v}</annotation>

</semantics>

</math></span><img src="./7c0375f8907febfc638d4311ee92fb289cf438f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.06ex; height:2.176ex;" alt="{\displaystyle w=u-v}" loading="lazy"></span> then, via the linearity of the equations, one finds that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w}</annotation>

</semantics>

</math></span><img src="./88b1e0c8e1be5ebe69d18a8010676fa42d7961e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}" loading="lazy"></span> satisfies

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}w_{t}&=Dw_{xx},&&0<x<1,\,t>0,\,D>0,\\w(x,0)&=0,\\w(0,t)&=0,\\w(1,t)&=0,\\\end{aligned}}}">

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}w_{t}&=Dw_{xx},&&0<x<1,\,t>0,\,D>0,\\w(x,0)&=0,\\w(0,t)&=0,\\w(1,t)&=0,\\\end{aligned}}}</annotation>

</semantics>

</math></span></span>

</p><p>Applying the energy estimate tells us <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|w(\cdot ,t)\|_{2}^{2}\leq 0}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|w(\cdot ,t)\|_{2}^{2}\leq 0}</annotation>

</semantics>

</math></span><img src="./632f4c4588a545b0cea1f1fdddc2e06bfbfaa7a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.634ex; height:3.176ex;" alt="{\displaystyle \|w(\cdot ,t)\|_{2}^{2}\leq 0}" loading="lazy"></span> which implies <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u=v}</annotation>

</semantics>

</math></span><img src="./2c14384063d79ad3532fd37aecc8616441da4631.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.556ex; height:1.676ex;" alt="{\displaystyle u=v}" loading="lazy"></span> (<a href="Almost_everywhere" title="Almost everywhere">almost everywhere</a>).

</p><p>Similarly, to show continuity with respect to initial conditions, assume that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

</semantics>

</math></span><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math></span><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"></span> are solutions corresponding to different initial data <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(x,0)=f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u(x,0)=f(x)}</annotation>

</semantics>

</math></span><img src="./ca093fdac61eb7caaac35db526d720decfa228c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.181ex; height:2.843ex;" alt="{\displaystyle u(x,0)=f(x)}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(x,0)=g(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v(x,0)=g(x)}</annotation>

</semantics>

</math></span><img src="./c27d35412447c4dd6c41d004ce9447370def68a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.816ex; height:2.843ex;" alt="{\displaystyle v(x,0)=g(x)}" loading="lazy"></span>.

Considering <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w=u-v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w=u-v}</annotation>

</semantics>

</math></span><img src="./7c0375f8907febfc638d4311ee92fb289cf438f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.06ex; height:2.176ex;" alt="{\displaystyle w=u-v}" loading="lazy"></span> once more, one finds that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w}</annotation>

</semantics>

</math></span><img src="./88b1e0c8e1be5ebe69d18a8010676fa42d7961e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}" loading="lazy"></span> satisfies the same equations as above but with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w(x,0)=f(x)-g(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w(x,0)=f(x)-g(x)}</annotation>

</semantics>

</math></span><img src="./5951fadb337b4d78e93f798d2212e12e94c19e22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.611ex; height:2.843ex;" alt="{\displaystyle w(x,0)=f(x)-g(x)}" loading="lazy"></span>.

This leads to the energy estimate <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|w(\cdot ,t)\|_{2}^{2}\leq D\|f(\cdot )-g(\cdot )\|_{2}^{2}}">

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|w(\cdot ,t)\|_{2}^{2}\leq D\|f(\cdot )-g(\cdot )\|_{2}^{2}}</annotation>

</semantics>

</math></span><img src="./bb6a4d1a6989f90358def36b3b923f52ff4deb90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.922ex; height:3.176ex;" alt="{\displaystyle \|w(\cdot ,t)\|_{2}^{2}\leq D\|f(\cdot )-g(\cdot )\|_{2}^{2}}" loading="lazy"></span> which establishes continuity (i.e. as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> become closer, as measured by the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L^{2}}</annotation>

</semantics>

</math></span><img src="./ba162c66ca85776c83557af5088cc6f8584d1912.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{2}}" loading="lazy"></span> norm of their difference, then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|w(\cdot ,t)\|_{2}\to 0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|w(\cdot ,t)\|_{2}\to 0}</annotation>

</semantics>

</math></span><img src="./fc242eca32b12fd1f0a325a21eaf3e8170ad7a07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.15ex; height:2.843ex;" alt="{\displaystyle \|w(\cdot ,t)\|_{2}\to 0}" loading="lazy"></span>).

</p><p>The <a href="Maximum_principle" title="Maximum principle">maximum principle</a> is an alternative approach to establish uniqueness and continuity of solutions with respect to initial conditions for this example.

The existence of solutions to this problem can be established using <a href="Heat_equation#Solving_the_heat_equation_using_Fourier_series" title="Heat equation">Fourier series</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Semi-group_theory">Semi-group theory</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="C0-semigroup" title="C0-semigroup">C0-semigroup</a></div>

<p>If it is possible to denote the solution to a Cauchy problem <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial u}{\partial t}}=Au,u(0)=u_{0}{\text{ (1)}}}">

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial u}{\partial t}}=Au,u(0)=u_{0}{\text{ (1)}}}</annotation>

</semantics>

</math></span><img src="./4b095576a87f36eac32100b33ce650f042142d7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:24.025ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial u}{\partial t}}=Au,u(0)=u_{0}{\text{ (1)}}}" loading="lazy"></span>, where <i>A</i> is a linear operator mapping a <a href="Dense_subspace" class="mw-redirect" title="Dense subspace">dense</a> linear subspace <i>D(A)</i> of <i>X</i> into <i>X</i> (see <a href="Discontinuous_linear_map" title="Discontinuous linear map">discontinuous linear map</a>), with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(t)=S(t)u_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u(t)=S(t)u_{0}}</annotation>

</semantics>

</math></span><img src="./c99501c7d98be7eb8ad57a1b6e76c43cc17f4d53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.609ex; height:2.843ex;" alt="{\displaystyle u(t)=S(t)u_{0}}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{S(t);t\geq 0\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{S(t);t\geq 0\}}</annotation>

</semantics>

</math></span><img src="./783e52b84acfa89db34e559096c9714f263276b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.608ex; height:2.843ex;" alt="{\displaystyle \{S(t);t\geq 0\}}" loading="lazy"></span> is a family of linear operators on <i>X</i>, satisfying

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(0)=I}</annotation>

</semantics>

</math></span><img src="./16d3fda2c3394fd938067837b56d2b6fe8bfeecb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.741ex; height:2.843ex;" alt="{\displaystyle S(0)=I}" loading="lazy"></span></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(a+b)=S(a)S(b)=S(b)S(a)}</annotation>

</semantics>

</math></span><img src="./878d2cda04f95b2491fa803a36c0d81dbb9a5085.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.262ex; height:2.843ex;" alt="{\displaystyle S(a+b)=S(a)S(b)=S(b)S(a)}" loading="lazy"></span> for any <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\geq 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a,b\geq 0}</annotation>

</semantics>

</math></span><img src="./9f3f5868710a6f6fa10bf8dd86011e6c85c40073.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.522ex; height:2.509ex;" alt="{\displaystyle a,b\geq 0}" loading="lazy"></span></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t\mapsto S(t)w}</annotation>

</semantics>

</math></span><img src="./0b442eb7e661cfd0fce56fa729ee0d0248fb27c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.266ex; height:2.843ex;" alt="{\displaystyle t\mapsto S(t)w}" loading="lazy"></span> is continuous for every <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w}</annotation>

</semantics>

</math></span><img src="./88b1e0c8e1be5ebe69d18a8010676fa42d7961e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}" loading="lazy"></span> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span></li>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}S(t)w=AS(t)w}</annotation>

</semantics>

</math></span><img src="./7648dd4c9dd0103897966e172a2b0e45c362ce28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.358ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dt}}S(t)w=AS(t)w}" loading="lazy"></span> for every <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w}</annotation>

</semantics>

</math></span><img src="./88b1e0c8e1be5ebe69d18a8010676fa42d7961e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}" loading="lazy"></span> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span></li></ul>

<p>then (1) is well-posed.

</p><p><a href="Hille%E2%80%93Yosida_theorem" title="Hille–Yosida theorem">Hille–Yosida theorem</a> states the criteria on <i>A</i> for such a <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{S(t);t\geq 0\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{S(t);t\geq 0\}}</annotation>

</semantics>

</math></span><img src="./783e52b84acfa89db34e559096c9714f263276b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.608ex; height:2.843ex;" alt="{\displaystyle \{S(t);t\geq 0\}}" loading="lazy"></span> to exist.

</p>

<ul><li><a href="Total_absorption_spectroscopy" title="Total absorption spectroscopy">Total absorption spectroscopy</a> – an example of an inverse problem or ill-posed problem in a real-life situation that is solved by means of the <a href="Expectation%E2%80%93maximization_algorithm" title="Expectation–maximization algorithm">expectation–maximization algorithm</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}

/* end https://en.wikipedia.org/ */

</style><div class="reflist reflist-lower-alpha">

<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">This definition of a well-posed problem comes from the work of <a href="Jacques_Hadamard" title="Jacques Hadamard">Jacques Hadamard</a> on <a href="Mathematical_model" title="Mathematical model">mathematical modeling</a> of <a href="Physical_phenomena" class="mw-redirect" title="Physical phenomena">physical phenomena</a>.</span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFHuang2019" class="citation journal cs1">Huang, Yunfei.; et al. (2019). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6345967">"Traction force microscopy with optimized regularization and automated Bayesian parameter selection for comparing cells"</a>. <i>Scientific Reports</i>. <b>9</b> (1): 537. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1810.05848">1810.05848</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2019NatSR...9..539H">2019NatSR...9..539H</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fs41598-018-36896-x">10.1038/s41598-018-36896-x</a></span>. <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6345967">6345967</a></span>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/30679578">30679578</a>.</cite></span>

</li>

</ol></div></div>

<ul><li><cite id="CITEREFHadamard1902" class="citation book cs1">Hadamard, Jacques (1902). <i>Sur les problèmes aux dérivées partielles et leur signification physique</i>. Princeton University Bulletin. pp. <span class="nowrap">49–</span>52.</cite></li>

<li><cite id="CITEREFParker1989" class="citation book cs1">Parker, Sybil B., ed. (1989) [1974]. <i>McGraw-Hill Dictionary of Scientific and Technical Terms</i> (4th ed.). New York: McGraw-Hill. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-07-045270-9</bdi>.</cite></li>

<li><cite id="CITEREFTikhonovArsenin1977" class="citation book cs1">Tikhonov, A. N.; Arsenin, V. Y. (1977). <i>Solutions of ill-Posed Problems</i>. New York: Winston. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-470-99124-0</bdi>.</cite></li>

<li><cite id="CITEREFStrauss2008" class="citation book cs1">Strauss, Walter A. (2008). <i>Partial differential equations; An introduction</i> (2nd ed.). Hoboken: Wiley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0470-05456-7</bdi>.</cite></li>

<li><cite id="CITEREFEvans1998" class="citation book cs1"><a href="Lawrence_C._Evans" title="Lawrence C. Evans">Evans, Lawrence C.</a> (1998). <a rel="nofollow" class="external text" href="https://math24.wordpress.com/wp-content/uploads/2013/02/partial-differential-equations-by-evans.pdf"><i>Partial differential equations</i></a> <span class="cs1-format">(PDF)</span>. Providence (R. I.): American mathematical society. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8218-0772-2</bdi>.</cite></li></ul>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}

/* end https://en.wikipedia.org/ */

</style><style data-mw-deduplicate="TemplateStyles:r1236075235">

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}

/* end https://en.wikipedia.org/ */

</style></div><div role="navigation" class="navbox authority-control" aria-label="Navbox391" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Authority control databases: National </th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4194555-4">Germany</a></span></li></ul></div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-06-25" href="https://en.wikipedia.org/wiki/?title=Well-posed_problem&oldid=1297345618">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>