SherbyNet - /page/wikibook/entry.mu

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<title>Weak formulation</title>

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Weak formulation

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Weak formulations are important tools for the analysis of mathematical <a href="Equation" title="Equation">equations</a> that permit the transfer of concepts of <a href="Linear_algebra" title="Linear algebra">linear algebra</a> to solve problems in other fields such as <a href="Partial_differential_equation" title="Partial differential equation">partial differential equations</a>. In a weak formulation, equations or conditions are no longer required to hold absolutely (and this is not even well defined) and has instead <a href="Weak_solution" title="Weak solution">weak solutions</a> only with respect to certain "test vectors" or "<a href="Test_function" class="mw-redirect" title="Test function">test functions</a>". In a strong formulation, the solution space is constructed such that these equations or conditions are already fulfilled.

The Lax–Milgram theorem, named after <a href="Peter_Lax" title="Peter Lax">Peter Lax</a> and <a href="Arthur_Milgram" title="Arthur Milgram">Arthur Milgram</a> who proved it in 1954, provides weak formulations for certain systems on <a href="Hilbert_space" title="Hilbert space">Hilbert spaces</a>.

<div class="mw-heading mw-heading2"><h2 id="General_concept">General concept</h2></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

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

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</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> be a <a href="Banach_space" title="Banach space">Banach space</a>, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V'}">

<msup>

</msup>

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

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</math><img src="./ff31fe992a31b4c954a933f1be91e2739a1c0ae7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.602ex; height:2.509ex;" alt="{\displaystyle V'}" loading="lazy"> be the <a href="Dual_space" title="Dual space">dual space</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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</mrow>

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

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</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\colon V\to V'}">

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</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle A\colon V\to V'}</annotation>

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</math><img src="./2ad9b64d338443b71ef561a91c30ced361b13d1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.78ex; height:2.509ex;" alt="{\displaystyle A\colon V\to V'}" loading="lazy"> be a <a href="Linear_map" title="Linear map">linear map</a>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in V'}">

<msup>

</msup>

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

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</math><img src="./2dde546b1ad7da7725268cd01b4766f860f09154.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.721ex; height:2.843ex;" alt="{\displaystyle f\in V'}" loading="lazy">.

A vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in V}">

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

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</math><img src="./636dd20088dea1139b38b3c04053ccf508bbed8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.958ex; height:2.176ex;" alt="{\displaystyle u\in V}" loading="lazy"> is a solution of the equation

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

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</math>

if and only if for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\in V}">

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</mrow>

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

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</math><img src="./99886ebbde63daa0224fb9bf56fa11b3c8a6f4fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.756ex; height:2.176ex;" alt="{\displaystyle v\in V}" loading="lazy">,

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

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A particular choice of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

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

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</math><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"> is called a test vector (in general) or a test function (if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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

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</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> is a function space).

To bring this into the generic form of a weak formulation, find <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in V}">

</mstyle>

</mrow>

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

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</math><img src="./636dd20088dea1139b38b3c04053ccf508bbed8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.958ex; height:2.176ex;" alt="{\displaystyle u\in V}" loading="lazy"> such that

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<annotation encoding="application/x-tex">{\displaystyle a(u,v)=f(v)\quad \forall v\in V,}</annotation>

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</math>

by defining the <a href="Bilinear_form" title="Bilinear form">bilinear form</a>

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<annotation encoding="application/x-tex">{\displaystyle a(u,v):=(Au)(v).}</annotation>

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</math>

<div class="mw-heading mw-heading2"><h2 id="Example_1:_linear_system_of_equations">Example 1: linear system of equations</h2></div>

Now, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=\mathbb {R} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle V=\mathbb {R} ^{n}}</annotation>

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</math><img src="./615f4ee928c0bb72f0a5d7a88580f7f5c739f06c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.782ex; height:2.343ex;" alt="{\displaystyle V=\mathbb {R} ^{n}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A:V\to V}">

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<annotation encoding="application/x-tex">{\displaystyle A:V\to V}</annotation>

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</math><img src="./2348ed6dd9b0f7cb231284ab34d02a6b60379ab8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.869ex; height:2.176ex;" alt="{\displaystyle A:V\to V}" loading="lazy"> be a <a href="Linear_mapping" class="mw-redirect" title="Linear mapping">linear mapping</a>. Then, the weak formulation of the equation

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

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involves finding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in V}">

</mstyle>

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

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</math><img src="./636dd20088dea1139b38b3c04053ccf508bbed8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.958ex; height:2.176ex;" alt="{\displaystyle u\in V}" loading="lazy"> such that for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\in V}">

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

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</math><img src="./99886ebbde63daa0224fb9bf56fa11b3c8a6f4fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.756ex; height:2.176ex;" alt="{\displaystyle v\in V}" loading="lazy"> the following equation holds:

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<annotation encoding="application/x-tex">{\displaystyle \langle Au,v\rangle =\langle f,v\rangle ,}</annotation>

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</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \cdot ,\cdot \rangle }">

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<annotation encoding="application/x-tex">{\displaystyle \langle \cdot ,\cdot \rangle }</annotation>

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</math><img src="./9a50080b735975d8001c9552ac2134b49ad534c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle \langle \cdot ,\cdot \rangle }" loading="lazy"> denotes an <a href="Inner_product" class="mw-redirect" title="Inner product">inner product</a>.

Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> is a linear mapping, it is sufficient to test with <a href="Basis_vector" class="mw-redirect" title="Basis vector">basis vectors</a>, and we get

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<annotation encoding="application/x-tex">{\displaystyle \langle Au,e_{i}\rangle =\langle f,e_{i}\rangle ,\quad i=1,\ldots ,n.}</annotation>

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</math>

Actually, expanding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=\sum _{j=1}^{n}u_{j}e_{j}}">

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</munderover>

<msub>

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</msub>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle u=\sum _{j=1}^{n}u_{j}e_{j}}</annotation>

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</math><img src="./4f16c2eedd38ea27eef248d698b15a76bec0f95d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:12.403ex; height:7.176ex;" alt="{\displaystyle u=\sum _{j=1}^{n}u_{j}e_{j}}" loading="lazy">, we obtain the <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> form of the equation

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

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</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{ij}=\langle Ae_{j},e_{i}\rangle }">

<msub>

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<msub>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle a_{ij}=\langle Ae_{j},e_{i}\rangle }</annotation>

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</math><img src="./cd9d610e740ebed7a9627452d7dd02d078a9a006.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.268ex; height:3.009ex;" alt="{\displaystyle a_{ij}=\langle Ae_{j},e_{i}\rangle }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}=\langle f,e_{i}\rangle }">

<msub>

</mrow>

</msub>

<msub>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f_{i}=\langle f,e_{i}\rangle }</annotation>

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</math><img src="./67e3997024d6387e7086ee1a119adf7106edf522.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.042ex; height:2.843ex;" alt="{\displaystyle f_{i}=\langle f,e_{i}\rangle }" loading="lazy">.

The bilinear form associated to this weak formulation is

<msup>

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<annotation encoding="application/x-tex">{\displaystyle a(u,v)=\mathbf {v} ^{T}\mathbf {A} \mathbf {u} .}</annotation>

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</math>

<div class="mw-heading mw-heading2"><h2 id="Example_2:_Poisson's_equation">Example 2: Poisson's equation</h2></div>

To solve <a href="Poisson's_equation" title="Poisson's equation">Poisson's equation</a>

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<annotation encoding="application/x-tex">{\displaystyle -\nabla ^{2}u=f,}</annotation>

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on a domain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega \subset \mathbb {R} ^{d}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega \subset \mathbb {R} ^{d}}</annotation>

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</math><img src="./f8c7db73e088aeed5bf0eff7d31ea5e1731bdaad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.547ex; height:2.676ex;" alt="{\displaystyle \Omega \subset \mathbb {R} ^{d}}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=0}">

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

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</math><img src="./41c2a269988ef0c55e0449b74950a4976e35a067.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle u=0}" loading="lazy"> on its <a href="Boundary_(topology)" title="Boundary (topology)">boundary</a>, and to specify the solution space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> later, one can use the <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>

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</math><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">-scalar product

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<annotation encoding="application/x-tex">{\displaystyle \langle u,v\rangle =\int _{\Omega }uv\,dx}</annotation>

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</math>

to derive the weak formulation. Then, testing with <a href="Differentiable_function" title="Differentiable function">differentiable functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

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

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</math><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"> yields

<msub>

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</msub>

<msup>

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</msup>

<msub>

</mrow>

</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle -\int _{\Omega }(\nabla ^{2}u)v\,dx=\int _{\Omega }fv\,dx.}</annotation>

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</math>

The left side of this equation can be made more symmetric by <a href="Integration_by_parts" title="Integration by parts">integration by parts</a> using <a href="Green's_identities" title="Green's identities">Green's identity</a> and assuming that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ba3d414a23bf4ecfa36cdd039241efc60a5bd9e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.389ex; height:2.176ex;" alt="{\displaystyle v=0}" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial \Omega }">

</mstyle>

</mrow>

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

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</math><img src="./16feddaad462c2a1c9efdaeee062a0484a023fde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.996ex; height:2.176ex;" alt="{\displaystyle \partial \Omega }" loading="lazy">:

<msub>

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</msub>

<msub>

</mrow>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \int _{\Omega }\nabla u\cdot \nabla v\,dx=\int _{\Omega }fv\,dx.}</annotation>

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</math>

This is what is usually called the weak formulation of <a href="Poisson's_equation" title="Poisson's equation">Poisson's equation</a>. <a href="Function_(mathematics)" title="Function (mathematics)">Functions</a> in the solution space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> must be zero on the boundary, and have square-integrable <a href="Derivative" title="Derivative">derivatives</a>. The appropriate space to satisfy these requirements is the <a href="Sobolev_space" title="Sobolev space">Sobolev space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{0}^{1}(\Omega )}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

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

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</math><img src="./830c3c449349317a46b77590a546b7af4a62dc08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.645ex; height:3.176ex;" alt="{\displaystyle H_{0}^{1}(\Omega )}" loading="lazy"> of functions with <a href="Weak_derivative" title="Weak derivative">weak derivatives</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}(\Omega )}">

<msup>

</mrow>

</msup>

</mstyle>

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

</semantics>

</math><img src="./2352f79f73ea92b82f762f072e41bb4a4cef2395.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.124ex; height:3.176ex;" alt="{\displaystyle L^{2}(\Omega )}" loading="lazy"> and with zero boundary conditions, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=H_{0}^{1}(\Omega )}">

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</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V=H_{0}^{1}(\Omega )}</annotation>

</semantics>

</math><img src="./3d18eb93030d20997ef696844fc3deb2a43bbd25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.531ex; height:3.176ex;" alt="{\displaystyle V=H_{0}^{1}(\Omega )}" loading="lazy">.

The generic form is obtained by assigning

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle a(u,v)=\int _{\Omega }\nabla u\cdot \nabla v\,dx}</annotation>

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</math>

and

<msub>

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</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f(v)=\int _{\Omega }fv\,dx.}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading2"><h2 id="The_Lax–Milgram_theorem">The Lax–Milgram theorem</h2></div>

This is a formulation of the Lax–Milgram theorem which relies on properties of the symmetric part of the <a href="Bilinear_form" title="Bilinear form">bilinear form</a>. It is not the most general form.

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> be a real <a href="Hilbert_space" title="Hilbert space">Hilbert space</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(\cdot ,\cdot )}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./fed8ed14a5035a5b274c792556b24b9fd6044cdd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.367ex; height:2.843ex;" alt="{\displaystyle a(\cdot ,\cdot )}" loading="lazy"> a <a href="Bilinear_form" title="Bilinear form">bilinear form</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">, which is

<ol><li><a href="Bilinear_form#On_normed_vector_spaces" title="Bilinear form">bounded</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |a(u,v)|\leq C\|u\|\|v\|\,;}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |a(u,v)|\leq C\|u\|\|v\|\,;}</annotation>

</semantics>

</math><img src="./214983b9fb61aa5ee503b29900877b384f20a025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.83ex; height:2.843ex;" alt="{\displaystyle |a(u,v)|\leq C\|u\|\|v\|\,;}" loading="lazy"> and</li>

<li><a href="Coercive_function#Coercive_operators_and_forms" title="Coercive function">coercive</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(u,u)\geq c\|u\|^{2}\,.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a(u,u)\geq c\|u\|^{2}\,.}</annotation>

</semantics>

</math><img src="./e790960c67d2d0b38803f2720bfed3a7f42c1d58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.58ex; height:3.176ex;" alt="{\displaystyle a(u,u)\geq c\|u\|^{2}\,.}" loading="lazy"></li></ol>

Then, for any bounded <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in V'}">

<msup>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2dde546b1ad7da7725268cd01b4766f860f09154.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.721ex; height:2.843ex;" alt="{\displaystyle f\in V'}" loading="lazy">, there is a unique solution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in V}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./636dd20088dea1139b38b3c04053ccf508bbed8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.958ex; height:2.176ex;" alt="{\displaystyle u\in V}" loading="lazy"> to the equation

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a(u,v)=f(v)\quad \forall v\in V}</annotation>

</semantics>

</math>

and it holds

<mfrac>

</mfrac>

</mrow>

<msub>

<msup>

</msup>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|u\|\leq {\frac {1}{c}}\|f\|_{V'}\,.}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading3"><h3 id="Application_to_example_1">Application to example 1</h3></div>

Here, application of the Lax–Milgram theorem is a stronger result than is needed.

<ul><li>Boundedness: all bilinear forms on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

</semantics>

</math><img src="./c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"> are bounded. In particular, we have <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |a(u,v)|\leq \|A\|\,\|u\|\,\|v\|}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |a(u,v)|\leq \|A\|\,\|u\|\,\|v\|}</annotation>

</semantics>

</math></li>

<li>Coercivity: this actually means that the <a href="Complex_number" title="Complex number">real parts</a> of the <a href="Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> are not smaller than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy">. Since this implies in particular that no eigenvalue is zero, the system is solvable.</li></ul>

Additionally, this yields the estimate

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|u\|\leq {\frac {1}{c}}\|f\|,}</annotation>

</semantics>

</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> is the minimal real part of an eigenvalue of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Application_to_example_2">Application to example 2</h3></div>

Here, choose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=H_{0}^{1}(\Omega )}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V=H_{0}^{1}(\Omega )}</annotation>

</semantics>

</math><img src="./3d18eb93030d20997ef696844fc3deb2a43bbd25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.531ex; height:3.176ex;" alt="{\displaystyle V=H_{0}^{1}(\Omega )}" loading="lazy"> with the norm

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|v\|_{V}:=\|\nabla v\|,}</annotation>

</semantics>

</math>

where the norm on the right is the <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><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">-norm on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"> (this provides a true norm on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> by the <a href="Poincar%C3%A9_inequality" title="Poincaré inequality">Poincaré inequality</a>).

But, we see that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |a(u,u)|=\|\nabla u\|^{2}}">

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |a(u,u)|=\|\nabla u\|^{2}}</annotation>

</semantics>

</math><img src="./a952d5b41c93b4446a54369ce76d39fa4b6d6352.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.769ex; height:3.176ex;" alt="{\displaystyle |a(u,u)|=\|\nabla u\|^{2}}" loading="lazy"> and by the <a href="Cauchy%E2%80%93Schwarz_inequality" title="Cauchy–Schwarz inequality">Cauchy–Schwarz inequality</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |a(u,v)|\leq \|\nabla u\|\,\|\nabla v\|}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |a(u,v)|\leq \|\nabla u\|\,\|\nabla v\|}</annotation>

</semantics>

</math><img src="./94bd7fae858776317dedb12fb3ef412f74258d03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.288ex; height:2.843ex;" alt="{\displaystyle |a(u,v)|\leq \|\nabla u\|\,\|\nabla v\|}" loading="lazy">.

Therefore, for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in [H_{0}^{1}(\Omega )]'}">

</mrow>

</mrow>

</msubsup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\in [H_{0}^{1}(\Omega )]'}</annotation>

</semantics>

</math><img src="./a5e949b3575a7b046830899b66fe9c45ae5515a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.743ex; height:3.176ex;" alt="{\displaystyle f\in [H_{0}^{1}(\Omega )]'}" loading="lazy">, there is a unique solution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in V}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./636dd20088dea1139b38b3c04053ccf508bbed8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.958ex; height:2.176ex;" alt="{\displaystyle u\in V}" loading="lazy"> of <a href="Poisson's_equation" title="Poisson's equation">Poisson's equation</a> and we have the estimate

<msub>

</mrow>

</mrow>

</msubsup>

<msup>

</msup>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \|\nabla u\|\leq \|f\|_{[H_{0}^{1}(\Omega )]'}.}</annotation>

</semantics>

</math>

<ul><li><a href="Babu%C5%A1ka%E2%80%93Lax%E2%80%93Milgram_theorem" title="Babuška–Lax–Milgram theorem">Babuška–Lax–Milgram theorem</a></li>

<li><a href="Lions%E2%80%93Lax%E2%80%93Milgram_theorem" title="Lions–Lax–Milgram theorem">Lions–Lax–Milgram theorem</a></li></ul>

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

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

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

</style><cite id="CITEREFLaxMilgram1954" class="citation cs2"><a href="Peter_Lax" title="Peter Lax">Lax, Peter D.</a>; <a href="Arthur_Milgram" title="Arthur Milgram">Milgram, Arthur N.</a> (1954), "Parabolic equations", Contributions to the theory of partial differential equations, Annals of Mathematics Studies, vol. 33, <a href="Princeton%2C_N._J." class="mw-redirect" title="Princeton, N. J.">Princeton, N. J.</a>: <a href="Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>, pp. 167–190, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2F9781400882182-010">10.1515/9781400882182-010</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781400882182</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0067317">0067317</a>, <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0058.08703">0058.08703</a></cite> <code class="cs1-code">{{citation}}</code>: ISBN / Date incompatibility (help)</li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Lax-MilgramTheorem.html">MathWorld page on Lax–Milgram theorem</a></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-01-02" href="https://en.wikipedia.org/wiki/?title=Weak_formulation&oldid=1266759545">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.

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