Lax-Milgram and Existence of Weak Solutions of Elliptic Boundary-Value Problems

Intuition Beginner

A differential equation asks you to find a function whose slopes and curvatures obey a rule at every single point. That is a stringent demand, and for rough data there may be no function smooth enough to satisfy it point by point. The trick of this unit is to stop testing the rule at points and start testing it on average, by multiplying the equation by an arbitrary "test" function and adding everything up. This converts the equation into a single balance condition: a certain weighted sum built from the unknown must match a weighted sum built from the data, for every test function you try. A function that passes all these averaged tests is called a weak solution.

Why trade the sharp point-by-point demand for a blurry averaged one? Because the averaged demand is far easier to satisfy, and once you find a weak solution you can often prove afterward that it was secretly smooth all along. The averaged form also hides a beautiful structure: the weighted sum on the left behaves like a tilted, bowl-shaped energy, and solving the equation is the same as finding the bottom of the bowl.

Here is the engine that finds that bottom. Picture an energy landscape over all candidate functions. Two features guarantee a unique lowest point. First, the landscape is bounded in steepness: nearby functions have nearby energies, so nothing jumps. Second, the landscape genuinely curves upward in every direction and never flattens into a valley floor that runs off to infinity; this upward curving is called coercivity. A steepness cap plus guaranteed upward curving, set over a space with no missing limit points, forces exactly one minimum to exist.

The Lax-Milgram theorem is this picture made into a guarantee. Whenever the averaged balance condition is bounded in steepness and curves upward, there is one and only one weak solution, and it depends continuously on the data: nudge the data a little and the solution moves only a little. For the elliptic equations of this unit, the upward-curving comes from the ellipticity of the operator, the precise statement that the equation diffuses in every direction rather than along a thin set of favored ones.

Visual Beginner

The single picture to hold is a tilted bowl-shaped energy over the space of candidate functions, with one lowest point, and the two guarantees that force that point to exist and be unique.

Read the three panels left to right. The left panel is the move from a point-by-point rule to a single averaged balance condition. Multiplying by a test function and summing turns the demanding local equation into a softer global one, tested against every test function at once.

The middle panel is the heart of the matter: the averaged balance defines a tilted bowl over the space of candidate functions. The cap on steepness means the bowl has no cliffs, and the upward curving means it has no flat-bottomed trenches running to infinity. A bowl with neither cliffs nor trenches, sitting over a space that contains all its limit points, has exactly one lowest point, and that point is the weak solution.

The right panel is continuous dependence. Tilt or reshape the bowl a little by changing the data, and its lowest point slides only a little. This is what makes weak solutions a usable notion: the solution is not just unique, it is stable against small perturbations of the data.

Worked example Beginner

We turn a simple boundary-value problem into its averaged form and check the upward-curving by hand. Take the interval from zero to one, and the equation $-u^{\prime \prime }(x)=f(x)$ with $u$ held at zero at both ends. This is the one-dimensional version of the elliptic problems of this unit.

Step 1. Multiply and sum. Pick any test function $v$ that is also zero at both ends. Multiply the equation by $v$ and add up over the interval: the left side becomes the total of $-u^{\prime \prime }(x)v(x)$ across the interval, and the right side becomes the total of $f(x)v(x)$.

Step 2. Move one slope across. Using integration by parts and the fact that $v$ vanishes at the two ends, the total of $-u^{\prime \prime }v$ equals the total of $u^{\prime }(x)v^{\prime }(x)$. So the averaged balance reads: the total of $u^{\prime }{v}^{\prime }$ equals the total of $fv$, for every test function $v$. The second derivative has been traded for a product of first slopes, which is the weak form.

Step 3. Read off the energy. Setting $v=u$ in the left side gives the total of $u^{\prime }(x)$ times itself, that is the total of $(u^{\prime }{)}^2$. This is the "bowl" value at $u$, and it can never be negative.

Step 4. Check the upward curving with a number. Take $u(x)=x(1-x)$, which is zero at both ends. Its slope is $u^{\prime }(x)=1-2x$. The total of $(u^{\prime }{)}^2$ over the interval works out to $1/3$, a strictly positive number. The energy of any nonzero clamped function is strictly positive, never zero; this is the upward curving that coercivity demands. By the Poincaré comparison of the sibling unit, the height of $u$ is also controlled by this same slope total, so the bowl really does climb in every direction.

Step 5. See what coercivity buys. Because the energy of a nonzero clamped function is strictly positive and bounded below by the function's own size, the bowl cannot run flat to infinity. A capped steepness plus this strictly positive curving give exactly one lowest point: one weak solution.

What this tells us: the weak form replaces a second-derivative equation by a balance of first slopes, and the positivity of the slope-energy is precisely the curving that guarantees a unique solution. The clamping at the ends is what makes the energy strictly positive, exactly as it made the Poincaré inequality possible.

Check your understanding Beginner

Formal definition Intermediate+

Throughout, $\Omega \subseteq {\mathbb{R}}^n$ is open and bounded, $H=H_0^1(\Omega )$ is the Hilbert space of 02.16.03 and 24.01.01 with inner product $⟨u,v{⟩}_{H_0^1}={\int }_{\Omega }Du\cdot Dvdx$ (equivalent, by the Poincaré inequality of 02.16.03, to the full $H^1$ inner product) and dual $H^{\ast }=H^{-1}(\Omega )$. The duality pairing of $f\in H^{\ast }$ with $v\in H$ is written $⟨f,v⟩$. All function spaces are real; the abstract theorem and its hypotheses are taken over a real Hilbert space, with 02.11.08 supplying completeness, the inner product, and the Riesz representation theorem.

Definition (bilinear form, boundedness, coercivity). Let $H$ be a real Hilbert space. A map $B:H\times H\to \mathbb{R}$ is a bilinear form if it is linear in each argument separately. It is bounded (continuous) if there is $\Lambda <\infty$ with $$|B(u,v)|\le \Lambda \Vert u{\Vert }_H\Vert v{\Vert }_H\text{for all }u,v\in H,$$ and **coercive** if there is $\beta >0$ with $$B(u,u)\ge \beta \Vert u{\Vert }_H^2\text{for all }u\in H.$$ No symmetry is assumed: $B(u,v)=B(v,u)$ is not required. When $B$ is symmetric, coercivity makes $B$ an equivalent inner product on $H$.

Definition (the elliptic operator and its weak form). Let $a^{ij},b^i,c\in L^{\infty }(\Omega )$ with $a^{ij}=a^{ji}$, and consider the second-order divergence-form operator $$Lu=-\sum _{i,j=1}^n{\partial }_i(a^{ij}(x){\partial }_{j}u)+\sum _{i=1}^n{b}^i(x){\partial }_{i}u+c(x)u.$$ $L$ is uniformly elliptic if there is $\theta >0$ with $$\sum _{i,j=1}^n{a}^{ij}(x){\xi }_i{\xi }_j\ge \theta |\xi {|}^2\text{for a.e. }x\in \Omega ,\text{all }\xi \in {\mathbb{R}}^n.$$ Multiplying $Lu=f$ by $v\in C_c^{\infty }(\Omega )$ and integrating by parts moves one derivative off the leading term and defines the bilinear form associated to $L$, $$B[u,v]={\int }_{\Omega }(\sum _{i,j}{a}^{ij}{\partial }_{j}u{\partial }_{i}v+\sum _i{b}^i({\partial }_{i}u)v+cuv)dx,$$ which extends to all $u,v\in H_0^1(\Omega )$ by density. Given $f\in H^{-1}(\Omega )$, a function $u\in H_0^1(\Omega )$ is a weak solution of the Dirichlet problem $Lu=f$ in $\Omega$, $u=0$ on $\partial \Omega$, if $$B[u,v]=⟨f,v⟩\text{for all }v\in H_0^1(\Omega ).$$ The boundary condition $u=0$ on $\partial \Omega$ is encoded by the membership $u\in H_0^1(\Omega )$ (zero trace, 02.16.03); no pointwise regularity of $u$ is presumed.

Counterexamples to common slips Intermediate+

• Coercivity is not automatic from ellipticity alone. Uniform ellipticity gives $\int a^{ij}{\partial }_{j}u{\partial }_{i}u\ge \theta \Vert Du{\Vert }_{L^2}^2$, but the lower-order terms can spoil positivity. With a large negative $c$, e.g. $Lu=-\Delta u-\kappa u$ for $\kappa$ above the first Dirichlet eigenvalue ${\lambda }_1$, the form $B[u,u]=\Vert Du{\Vert }_{L^2}^2-\kappa \Vert u{\Vert }_{L^2}^2$ is negative on the first eigenfunction. Lax-Milgram does not apply; one shifts to $L+\mu$ and uses the Fredholm alternative instead.

• Boundedness needs the coefficients in $L^{\infty }$, not merely measurable. If some $a^{ij}$ is unbounded, $B$ need not satisfy $|B[u,v]|\le \Lambda \Vert u{\Vert }_{H_0^1}\Vert v{\Vert }_{H_0^1}$, and the form is not even defined on all of $H_0^1$. The $L^{\infty }$ hypothesis is what makes $B$ a bounded bilinear form on the Sobolev space.

• Symmetry is a convenience, not a requirement. The first-order drift $b^i{\partial }_{i}u$ makes $B$ non-symmetric. The symmetric Riesz/variational route (minimizing $\frac{1}{2}B[u,u]-⟨f,u⟩$) then fails, since a non-symmetric $B$ is not the second differential of an energy. Lax-Milgram is exactly the extension of Riesz representation that survives the loss of symmetry.

• The wrong sign of the leading term destroys coercivity. Writing $Lu=+{\partial }_i(a^{ij}{\partial }_{j}u)+\cdots$ (the backward heat / anti-elliptic sign) flips the leading quadratic to $-\theta \Vert Du{\Vert }^2$, and no coercivity constant $\beta >0$ exists. The divergence-form minus sign is the convention that makes the principal part positive; this unit fixes that sign in the definition of $L$.

Key theorem with proof Intermediate+

Theorem (Lax-Milgram). Let $H$ be a real Hilbert space and $B:H\times H\to \mathbb{R}$ a bounded coercive bilinear form, with constants $\Lambda$ and $\beta$ as above. Then for every bounded linear functional $f\in H^{\ast }$ there is a unique $u\in H$ with $$B[u,v]=⟨f,v⟩\text{for all }v\in H,$$ and the solution obeys the a priori bound $\Vert u{\Vert }_H\le {\beta }^{-1}\Vert f{\Vert }_{H^{\ast }}$ ^{[Lax-Milgram 1954] [Evans 2010 §6.2]}.

Proof. For each fixed $u\in H$, the map $v\mapsto B[u,v]$ is a bounded linear functional on $H$, of norm at most $\Lambda \Vert u{\Vert }_H$. By the Riesz representation theorem 02.11.08 there is a unique element, call it $Au\in H$, with $B[u,v]=⟨Au,v{⟩}_H$ for all $v$. The map $A:H\to H$ is linear (bilinearity of $B$ and uniqueness of the Riesz representative) and bounded: $\Vert Au{\Vert }_H={\sup }_{\Vert v{\Vert }_H=1}⟨Au,v{⟩}_H={\sup }_{\Vert v\Vert =1}B[u,v]\le \Lambda \Vert u{\Vert }_H$. Likewise $f\in H^{\ast }$ has a Riesz representative $w\in H$ with $⟨f,v⟩=⟨w,v{⟩}_H$. The problem $B[u,v]=⟨f,v⟩$ for all $v$ is therefore equivalent to the single equation $Au=w$ in $H$.

It remains to show $A$ is a bijection of $H$. Coercivity gives, for every $u$, $$\beta \Vert u{\Vert }_H^2\le B[u,u]=⟨Au,u{⟩}_H\le \Vert Au{\Vert }_H\Vert u{\Vert }_H,$$ so $\Vert Au{\Vert }_H\ge \beta \Vert u{\Vert }_H$. This lower bound forces $A$ to be injective and to have closed range: if $A{u}_k\to g$, then $(u_k)$ is Cauchy because $\Vert u_k-u_m{\Vert }_H\le {\beta }^{-1}\Vert A{u}_k-A{u}_m{\Vert }_H$, so $u_k\to u$ and $Au=g$ by continuity. Suppose the range $R(A)$ were a proper closed subspace; then there is a nonzero $z\perp R(A)$, and in particular $\beta \Vert z{\Vert }_H^2\le B[z,z]=⟨Az,z{⟩}_H=0$, forcing $z=0$, a contradiction. Hence $R(A)=H$ and $A$ is a continuous linear bijection with $\Vert A^{-1}\Vert \le {\beta }^{-1}$. Setting $u=A^{-1}w$ solves $Au=w$ uniquely, and $\Vert u{\Vert }_H=\Vert A^{-1}w{\Vert }_H\le {\beta }^{-1}\Vert w{\Vert }_H={\beta }^{-1}\Vert f{\Vert }_{H^{\ast }}$. $\square$

An equivalent and constructive proof replaces the closed-range argument by a contraction. For $\rho >0$ the affine map $T_{\rho }v=v-\rho (Av-w)$ satisfies $\Vert T_{\rho }{v}_1-T_{\rho }{v}_2{\Vert }_H^2=\Vert v{\Vert }_H^2-2\rho ⟨Av,v{⟩}_H+{\rho }^2\Vert Av{\Vert }_H^2\le (1-2\rho \beta +{\rho }^2{\Lambda }^2)\Vert v{\Vert }_H^2$ with $v=v_1-v_2$; choosing $\rho \in (0,2\beta /{\Lambda }^2)$ makes the bracket strictly below $1$, so $T_{\rho }$ is a contraction on the complete space $H$ and the Banach fixed-point theorem yields the unique fixed point $u=T_{\rho }u$, i.e. $Au=w$.

Bridge. Lax-Milgram is exactly the Riesz representation theorem of 02.11.08 with symmetry removed: when $B$ is symmetric and coercive it is an equivalent inner product and the solution is the genuine Riesz representative of $f$, while the general case substitutes the bounded inverse $A^{-1}$, built either by the closed-range argument or by the contraction, for that representative — this is the foundational reason existence here costs no compactness and no spectral theory. The a priori bound $\Vert u{\Vert }_H\le {\beta }^{-1}\Vert f{\Vert }_{H^{\ast }}$ is the continuous-dependence statement of the Beginner tier made quantitative, and it builds toward the energy method for elliptic problems where coercivity is supplied by uniform ellipticity through Gårding's inequality. The abstract theorem appears again in 24.01.03 in the finite-dimensional Galerkin form that underlies the finite element method, where the same coercivity gives Céa's quasi-optimality estimate; putting these together, the central insight is that a bounded coercive bilinear form on a complete space is invertible by representation alone, and this is exactly what turns the averaged weak formulation into a solvable problem.

Exercises Intermediate+

Advanced results Master

The abstract theorem and the elliptic existence theorem sit inside a larger structure: the variational inequalities of Stampacchia that generalize the equation to convex constraints, the Babuška-Nečas inf-sup condition that replaces coercivity for saddle-point and non-coercive problems, the Gårding-plus-Fredholm dichotomy that handles the full range of zeroth-order coefficients, the Galerkin discretization that descends the whole apparatus to finite dimensions, and the higher regularity that recovers the classical solution from the weak one. Each refines the bounded-coercive-form argument of the Intermediate tier.

Theorem 1 (Stampacchia, variational inequalities). Let $B$ be a bounded coercive bilinear form on $H$ and $C\subseteq H$ a nonempty closed convex set. For every $f\in H^{\ast }$ there is a unique $u\in C$ with $$B[u,v-u]\ge ⟨f,v-u⟩\text{for all }v\in C,$$ and $u$ depends Lipschitz-continuously on $f$ ^{[Stampacchia 1964]}. When $C=H$ the inequality forces equality and recovers Lax-Milgram; when $B$ is symmetric, $u$ minimizes $\frac{1}{2}B[v,v]-⟨f,v⟩$ over $C$, the constrained Dirichlet principle. The proof iterates the projection onto $C$ (which exists and is a contraction because $C$ is closed convex in a Hilbert space) with the contraction $T_{\rho }$ of the Lax-Milgram fixed-point argument; the composition is still a contraction for small $\rho$, and its fixed point solves the inequality. Obstacle problems and elliptic free-boundary problems are the standard instances.

Theorem 2 (Babuška-Nečas inf-sup; the coercivity-free generalization). Coercivity is sufficient but not necessary for well-posedness of $B[u,v]=⟨f,v⟩$. On Hilbert spaces $U$ (trial) and $V$ (test), the problem is well-posed for every $f\in V^{\ast }$ if and only if $B$ is bounded, satisfies the inf-sup condition $$\underset{u\in U\setminus \{0\}}{\inf }\underset{v\in V\setminus \{0\}}{\sup }\frac{B[u,v]}{\Vert u{\Vert }_U\Vert v{\Vert }_V}=\alpha >0,$$ and is non-degenerate in the test variable (${\sup }_{u}B[u,v]>0$ for each $v\ne 0$), with $\Vert u{\Vert }_U\le {\alpha }^{-1}\Vert f{\Vert }_{V^{\ast }}$. Coercivity is the special case $U=V$ with the diagonal $u=v$ already realizing the supremum. The inf-sup framework is what makes mixed finite element methods (Stokes flow, Darcy flow, the Hellinger-Reissner elasticity formulation) well-posed, where the natural bilinear form is a saddle-point form with no coercivity on the whole space.

Theorem 3 (the full Fredholm alternative for $L$). Let $L$ be uniformly elliptic with $L^{\infty }$ coefficients on a bounded domain. There is ${\mu }_0\ge 0$ (the Gårding constant) such that for $\mu \ge {\mu }_0$, $L+\mu$ is boundedly invertible $H^{-1}\to H_0^1$. Consequently exactly one of the following holds ^{[Evans 2010 §6.2] [Gilbarg-Trudinger 1983 §8.6]}: (i) for every $f\in H^{-1}$ the problem $Lu=f$ has a unique weak solution; or (ii) the homogeneous problem $Lu=0$ has a nonzero solution, the solution spaces of $Lu=0$ and of the formal adjoint $L^{\ast }w=0$ are finite-dimensional of equal dimension, and $Lu=f$ is solvable iff $f$ annihilates that adjoint kernel. The set of $\lambda$ for which $Lu=\lambda u$ has a nonzero solution — the spectrum — is at most countable, real when $L$ is self-adjoint, and discrete with no finite accumulation point, by the compactness of $H_0^1\hookrightarrow \hookrightarrow L^2$ 02.16.03 applied to the resolvent. This is the elliptic analogue of the finite-dimensional dichotomy "a square matrix is either invertible or has a kernel."

Theorem 4 (Galerkin approximation and Céa's lemma). Let $V_h\subseteq H$ be a finite-dimensional subspace and $u_h\in V_h$ the unique solution of $B[u_h,v_h]=⟨f,v_h⟩$ for all $v_h\in V_h$ (existence by Lax-Milgram applied on $V_h$). Then the Galerkin solution is quasi-optimal: $$\Vert u-u_h{\Vert }_H\le \frac{\Lambda }{\beta }\underset{v_h\in V_h}{\inf }\Vert u-v_h{\Vert }_H,$$ so the discrete error is, up to the condition-number factor $\Lambda /\beta$, the best approximation error of $u$ from $V_h$. The proof is Galerkin orthogonality $B[u-u_h,v_h]=0$ combined with coercivity and boundedness. This is the theoretical foundation of the finite element method of 24.01.03: convergence of the numerical scheme is reduced to the approximation power of the subspaces $V_h$.

Theorem 5 (interior $H^2$ regularity). If additionally $a^{ij}\in C^1(\Omega )$, $b^i,c\in L^{\infty }$, and $f\in L^2(\Omega )$, then the weak solution $u\in H_0^1(\Omega )$ of $Lu=f$ lies in $H_{\mathrm{loc}}^2(\Omega )$ with the estimate $\Vert u{\Vert }_{H^2({\Omega }^{\prime })}\le C(\Vert f{\Vert }_{L^2(\Omega )}+\Vert u{\Vert }_{L^2(\Omega )})$ for ${\Omega }^{\prime }⋐\Omega$, and $Lu=f$ holds a.e. ^{[Evans 2010 §6.3] [Gilbarg-Trudinger 1983 §8.3]}. The method is difference quotients: the translation estimate $\Vert {\tau }_{h}u-u{\Vert }_{L^2}\le |h|\Vert Du{\Vert }_{L^2}$ of 02.16.03 is run in reverse — a uniform bound on the difference quotients $D_k^{h}u$ in $H^1$ promotes $Du$ to $H^1$, hence $u$ to $H^2$. Bootstrapping with Schauder or $L^p$ theory recovers the classical solution when the data are smooth: the weak solution found by Lax-Milgram was the classical solution all along.

Synthesis. Lax-Milgram is the foundational reason the weak formulation is solvable: it is exactly the Riesz representation theorem with symmetry removed, and the existence of weak solutions of elliptic boundary-value problems is this single abstract fact fed the bilinear form whose coercivity comes from uniform ellipticity through Gårding's inequality. The a priori bound $\Vert u{\Vert }_{H_0^1}\le {\beta }^{-1}\Vert f{\Vert }_{H^{-1}}$ is dual to the coercivity constant, one read as stability of the solution and the other as positivity of the energy, and putting these together the central insight is that bounded-plus-coercive is the entire content of well-posedness in the coercive case — generalised by Babuška-Nečas to the inf-sup condition when the diagonal $u=v$ no longer realizes the supremum, and by Stampacchia to convex constraints when the equation becomes a variational inequality.

The compactness of the Sobolev embedding 02.16.03 is what upgrades the coercive shift $L+\mu$ into the Fredholm alternative for the full operator, so the existence theory and the spectral theory of $L$ are the same compactness read twice; this is exactly the structure that descends, through Galerkin orthogonality and Céa's quasi-optimality, to the finite element method of 24.01.03, where the same constants $\Lambda$ and $\beta$ that proved existence now control the numerical error. The whole edifice appears again wherever an elliptic problem must be solved before it can be analyzed: the higher regularity of Theorem 5 recovers the classical solution from the weak one by running the translation estimate of 02.16.03 in reverse, closing the loop from the rough averaged formulation back to the smooth point-by-point equation.

Full proof set Master

Proposition 1 (existence and uniqueness for coercive $L$). Let $L$ be uniformly elliptic on a bounded $\Omega$ with $a^{ij},b^i,c\in L^{\infty }$, and suppose the associated form $B$ is coercive on $H_0^1(\Omega )$ (e.g. $b\equiv 0$, $c\ge 0$, or $c$ above the threshold of Exercise 5). Then for every $f\in H^{-1}(\Omega )$ there is a unique weak solution $u\in H_0^1(\Omega )$ of $Lu=f$, with $\Vert u{\Vert }_{H_0^1}\le {\beta }^{-1}\Vert f{\Vert }_{H^{-1}}$.

Proof. Boundedness of $B$ on $H_0^1$ is the estimate of Exercise 3, valid because the coefficients lie in $L^{\infty }$ and the Poincaré inequality of 02.16.03 controls $\Vert u{\Vert }_{L^2}$ by $\Vert u{\Vert }_{H_0^1}$. Coercivity is assumed. The functional $v\mapsto ⟨f,v⟩$ is bounded on $H_0^1$ by definition of $H^{-1}$. The Lax-Milgram theorem applied to the real Hilbert space $H=H_0^1(\Omega )$ (complete by 24.01.01, with inner product and Riesz representation from 02.11.08) yields a unique $u$ with $B[u,v]=⟨f,v⟩$ for all $v\in H_0^1$, which is the definition of a weak solution, and the a priori bound is the Lax-Milgram estimate. $\square$

Proposition 2 (Gårding's inequality). Under uniform ellipticity with constant $\theta$ and coefficients in $L^{\infty }$, there are $\beta =\theta /2>0$ and $\gamma \ge 0$ with $B[u,u]\ge \beta \Vert u{\Vert }_{H_0^1}^2-\gamma \Vert u{\Vert }_{L^2}^2$ for all $u\in H_0^1(\Omega )$.

Proof. Ellipticity gives ${\int }_{\Omega }{a}^{ij}{\partial }_{j}u{\partial }_{i}u\ge \theta {\int }_{\Omega }|Du{|}^2=\theta \Vert u{\Vert }_{H_0^1}^2$. For the first-order term, Cauchy-Schwarz and Young's inequality with parameter $ϵ>0$ give $|\int b^i({\partial }_{i}u)u|\le \Vert b{\Vert }_{L^{\infty }}\sqrt{n}\Vert Du{\Vert }_{L^2}\Vert u{\Vert }_{L^2}\le \frac{\theta }{2}\Vert Du{\Vert }_{L^2}^2+\frac{n\Vert b{\Vert }_{L^{\infty }}^2}{2\theta }\Vert u{\Vert }_{L^2}^2$, choosing $ϵ$ so the first term is $\frac{\theta }{2}\Vert Du{\Vert }^2$. The zeroth-order term satisfies $\int c{u}^2\ge -\Vert c{\Vert }_{L^{\infty }}\Vert u{\Vert }_{L^2}^2$. Summing the three contributions, $$B[u,u]\ge \theta \Vert u{\Vert }_{H_0^1}^2-\frac{\theta }{2}\Vert u{\Vert }_{H_0^1}^2-(\frac{n\Vert b{\Vert }_{L^{\infty }}^2}{2\theta }+\Vert c{\Vert }_{L^{\infty }})\Vert u{\Vert }_{L^2}^2,$$ which is the claim with $\beta =\theta /2$ and $\gamma =n\Vert b{\Vert }_{L^{\infty }}^2/(2\theta )+\Vert c{\Vert }_{L^{\infty }}$. When $b\equiv 0$ and $c\ge 0$ both subtracted terms vanish and $B$ is coercive outright. $\square$

Proposition 3 (solvability of the shifted problem and the Fredholm alternative). With $\gamma$ as in Proposition 2, $L+\mu$ has a coercive form for every $\mu \ge \gamma$, so $(L+\mu {)}^{-1}=:G$ is a bounded operator $H^{-1}\to H_0^1$; moreover $G:L^2\to L^2$ is compact, and the Fredholm alternative holds for $Lu=f$.

Proof. For $\mu \ge \gamma$, $B_{\mu }[u,u]=B[u,u]+\mu \Vert u{\Vert }_{L^2}^2\ge \beta \Vert u{\Vert }_{H_0^1}^2+(\mu -\gamma )\Vert u{\Vert }_{L^2}^2\ge \beta \Vert u{\Vert }_{H_0^1}^2$, and $B_{\mu }$ is bounded, so by Proposition 1 the map $G:g\mapsto u$ solving $(L+\mu )u=g$ is well-defined and bounded $H^{-1}\to H_0^1$. Restricting the source to $L^2\hookrightarrow H^{-1}$ and post-composing with the compact embedding $H_0^1\hookrightarrow \hookrightarrow L^2$ of 02.16.03, $G:L^2\to L^2$ is the composition of a bounded and a compact map, hence compact. A function $u$ solves $Lu=f$ iff $(L+\mu )u=f+\mu u$, i.e. $u=G(f+\mu u)$, i.e. $(I-\mu G)u=Gf$ in $L^2$. Since $\mu G$ is compact and self-adjoint when $L$ is (positive coercive symmetric form), the Fredholm alternative and spectral theorem for compact operators 02.11.08 give: either $I-\mu G$ is invertible and $Lu=f$ has a unique solution for all $f$, or $1$ is an eigenvalue of $\mu G$ of finite multiplicity, the kernels of $L$ and $L^{\ast }$ have equal finite dimension, and $Lu=f$ is solvable iff $f\perp \ker L^{\ast }$. The eigenvalues of $\mu G$ accumulate only at $0$, so the spectrum of $L$ is discrete with no finite accumulation point. $\square$

Proposition 4 (Céa quasi-optimality of the Galerkin solution). Let $V_h\subseteq H_0^1(\Omega )$ be finite-dimensional and $u_h$ the Galerkin solution. Then $\Vert u-u_h{\Vert }_{H_0^1}\le (\Lambda /\beta ){\inf }_{v_h\in V_h}\Vert u-v_h{\Vert }_{H_0^1}$.

Proof. Lax-Milgram on the finite-dimensional (hence complete) subspace $V_h$ gives existence and uniqueness of $u_h$ with $B[u_h,v_h]=⟨f,v_h⟩$ for all $v_h\in V_h$. Subtracting from $B[u,v_h]=⟨f,v_h⟩$ yields Galerkin orthogonality $B[u-u_h,v_h]=0$ for all $v_h\in V_h$. For any $v_h\in V_h$, coercivity and orthogonality give $\beta \Vert u-u_h{\Vert }_{H_0^1}^2\le B[u-u_h,u-u_h]=B[u-u_h,u-v_h]\le \Lambda \Vert u-u_h{\Vert }_{H_0^1}\Vert u-v_h{\Vert }_{H_0^1}$, where the middle equality inserts $v_h-u_h\in V_h$ into the orthogonal slot. Dividing by $\beta \Vert u-u_h{\Vert }_{H_0^1}$ gives $\Vert u-u_h{\Vert }_{H_0^1}\le (\Lambda /\beta )\Vert u-v_h{\Vert }_{H_0^1}$, and taking the infimum over $v_h$ completes the proof. $\square$

Connections Master

• The abstract engine is the Riesz representation theorem and the Hilbert-space structure of 02.11.08: Lax-Milgram is that theorem with symmetry dropped, and the whole existence theory of this unit is the representation of the data functional $f$ by a solution $u$ through the bounded inverse of the form-induced operator $A$. This unit owns the non-symmetric extension and its elliptic application; 02.11.08 owns the symmetric inner-product representation.

• The coercivity that powers the energy method is the Poincaré inequality and the compact embedding of 02.16.03: Poincaré makes $\Vert Du{\Vert }_{L^2}$ an equivalent norm so that uniform ellipticity yields coercivity, and the compactness $H_0^1\hookrightarrow \hookrightarrow L^2$ is exactly what turns the coercive shift $L+\mu$ into the Fredholm alternative and the discrete spectrum for the full operator. The sibling unit supplies both the coercivity input and the compactness input to this existence theory.

• The Sobolev-space framework — the spaces $H_0^1$ and $H^{-1}$, weak derivatives, the trace giving meaning to the zero boundary condition — is built in 24.01.01, and the weak/variational formulation itself, together with the finite element discretization governed by Céa's lemma, is developed concretely for the Poisson case in 24.01.03. This unit provides the general-operator existence theorem of which 24.01.03 treats the symmetric Poisson special case.

• The higher regularity that recovers a classical solution from the weak one runs the difference-quotient method, the reverse of the translation estimate $\Vert {\tau }_{h}u-u{\Vert }_{L^2}\le |h|\Vert Du{\Vert }_{L^2}$ proved in 02.16.03; the Schauder and De Giorgi-Nash-Moser theories of 02.17.04 and 02.17.07 then bootstrap the weak solution to full classical regularity, so the existence theorem of this unit is the entry point to the regularity chapter.

• The Fredholm alternative and the discreteness of the elliptic spectrum invoke the compact-operator and spectral theory of 02.11.05: the solution operator $G$ of the coercive shift composed with the compact Sobolev embedding is the single compact operator whose Fredholm theory governs solvability of $Lu=f$, making this unit the place where abstract compact-operator theory meets the concrete elliptic boundary-value problem.

Historical & philosophical context Master

The weak formulation of elliptic problems grew out of the Dirichlet principle, the nineteenth-century idea — championed by Riemann and rehabilitated by Hilbert — of solving $-\Delta u=f$ by minimizing the Dirichlet energy. The rigorous Hilbert-space form of the existence question crystallized after Frigyes Riesz's 1934 representation theorem for continuous linear functionals on a Hilbert space ^{[Riesz 1934]}, which identified the symmetric case completely. Peter Lax and Arthur Milgram, in their 1954 paper in the Contributions to the Theory of Partial Differential Equations volume of the Annals of Mathematics Studies ^{[Lax-Milgram 1954]}, extended representation to bounded coercive forms without symmetry, precisely the generality needed for elliptic operators carrying first-order drift; their motivation was parabolic equations, where the non-self-adjoint structure is unavoidable.

The coercivity that makes the method work for genuine elliptic operators is Lars Gårding's 1953 inequality ^{[Gårding 1953]}, which derived the energy estimate $B[u,u]\ge \beta \Vert u{\Vert }_{H_0^1}^2-\gamma \Vert u{\Vert }_{L^2}^2$ from uniform ellipticity and thereby connected the algebraic positivity of the coefficient matrix to the functional-analytic coercivity of the form. The Fredholm alternative invoked for the non-coercive range traces to Erik Ivar Fredholm's 1903 Acta Mathematica theory of integral equations ^{[Fredholm 1903]}, whose dichotomy for $I-K$ with $K$ compact is the abstract template realized here through the compact Sobolev embedding. Guido Stampacchia's 1964 Comptes Rendus note ^{[Stampacchia 1964]} extended the existence theory from equations to variational inequalities on convex sets, opening the modern theory of obstacle and free-boundary problems, and the inf-sup condition formalized by Ivo Babuška and Jindřich Nečas removed even coercivity, supplying the well-posedness theory for the saddle-point problems of mixed finite elements.

Bibliography Master

@incollection{LaxMilgram1954,
  author    = {Lax, Peter D. and Milgram, Arthur N.},
  title     = {Parabolic equations},
  booktitle = {Contributions to the Theory of Partial Differential Equations},
  series    = {Annals of Mathematics Studies},
  number    = {33},
  publisher = {Princeton University Press},
  year      = {1954},
  pages     = {167--190}
}

@article{Garding1953,
  author  = {G\aa{}rding, Lars},
  title   = {Dirichlet's problem for linear elliptic partial differential equations},
  journal = {Mathematica Scandinavica},
  volume  = {1},
  year    = {1953},
  pages   = {55--72}
}

@article{Stampacchia1964,
  author  = {Stampacchia, Guido},
  title   = {Formes bilin\'eaires coercitives sur les ensembles convexes},
  journal = {Comptes Rendus de l'Acad\'emie des Sciences Paris},
  volume  = {258},
  year    = {1964},
  pages   = {4413--4416}
}

@article{Fredholm1903,
  author  = {Fredholm, Erik Ivar},
  title   = {Sur une classe d'\'equations fonctionnelles},
  journal = {Acta Mathematica},
  volume  = {27},
  year    = {1903},
  pages   = {365--390}
}

@article{Riesz1934,
  author  = {Riesz, Frigyes},
  title   = {Zur Theorie des Hilbertschen Raumes},
  journal = {Acta Scientiarum Mathematicarum (Szeged)},
  volume  = {7},
  year    = {1934},
  pages   = {34--38}
}

@book{Necas2012,
  author    = {Ne\v{c}as, Ji\v{r}\'i},
  title     = {Direct Methods in the Theory of Elliptic Equations},
  series    = {Springer Monographs in Mathematics},
  publisher = {Springer},
  year      = {2012}
}