Application: Weak Formulation of PDEs

From classical to weak solutions¶

Fundamental solutions give explicit formulas, but only for constant-coefficient operators on all of $\mathbb{R}^d$. For bounded domains with boundary conditions, or for variable-coefficient operators, we need a different approach. Sobolev spaces provide it: instead of finding a formula, we prove existence abstractly using duality.

The idea is to rewrite the PDE as a problem about functionals on a Hilbert space. Consider $-\Delta u = f$ on a bounded domain $\Omega$ with $u = 0$ on $\partial\Omega$. The boundary condition is encoded by working in $H^1_0(\Omega)$ (the closure of $C_c^\infty(\Omega)$ in $H^1$). Multiply the PDE by a test function $v \in H^1_0(\Omega)$ and integrate by parts:

The boundary terms vanish because $v \in H^1_0$. This is the weak formulation: find $u \in H^1_0(\Omega)$ satisfying the identity above.

Existence by the Riesz representation theorem¶

The PDE is an equation in the dual space. The weak formulation is not an equation between functions in $\Omega$; it is an equation between elements of $H^{-1}(\Omega) = H^1_0(\Omega)^*$. Read both sides of $(\text{weak})$ as functionals of the test direction $v$:

The left-hand functional $Au$ is continuous on $H^1_0$ by Cauchy-Schwarz ($|Au(v)| \leq \|\nabla u\|_{L^2}\|\nabla v\|_{L^2}$), so $A : H^1_0 \to H^{-1}$ is a bounded operator. The right-hand functional $F$ is continuous on $H^1_0$ by Hölder plus Poincaré (for $f \in L^2$, or more generally for $f \in H^{-1}$ by definition). Solving the PDE means solving the linear equation $Au = F$ in $H^{-1}$.

This is the shift that distributions made possible. Classically $-\Delta u = f$ is an equation between functions at each point $x \in \Omega$; weakly, it is a single equation between two bounded linear functionals on $H^1_0$. The equality is tested by pairing against every $v \in H^1_0$, not by comparing pointwise values.

The left-hand side is simultaneously an inner product on $H^1_0(\Omega)$, $\langle u, v \rangle_{\dot{H}^1} = \int \nabla u \cdot \nabla v\, dx$ (using the Poincaré inequality to show $\|\nabla \cdot\|_{L^2}$ is a norm equivalent to $\|\cdot\|_{H^1}$ on $H^1_0$). That is what makes Riesz applicable: the operator $A$ is the Riesz isomorphism $H^1_0 \xrightarrow{\sim} H^{-1}$ associated with this inner product.

By the Riesz representation theorem (Theorem 1), there exists a unique $u_f \in H^1_0(\Omega)$ such that

This $u_f$ is the weak solution. Existence and uniqueness are immediate corollaries of a theorem we already proved in the duality chapter.

The Lax-Milgram theorem¶

The Riesz argument above works because the left-hand side $\int \nabla u \cdot \nabla v\, dx$ is an inner product on $H^1_0$. For more general operators (with lower-order terms, non-symmetric coefficients, or convection) the bilinear form $B(u,v)$ may not be symmetric, so it is not an inner product. The Lax-Milgram theorem handles this case.

The Gelfand triple¶

The natural framework for weak formulations is a triple of spaces

where $V$ is a Hilbert space (the “energy space”), $H$ is a pivot space, and $V^*$ is the dual. The embeddings are continuous and dense.

The prototypical example is

An element $f \in H^{-1}$ acts on test functions in $H^1_0$ via the duality pairing. The PDE $Au = f$ is an equation in $H^{-1}$: we seek $u \in H^1_0$ such that $B(u,v) = \langle f, v \rangle_{H^{-1} \times H^1_0}$ for all $v \in H^1_0$. The Gelfand triple identifies which space each player lives in:

• The solution $u$ lives in the energy space $V = H^1_0$.

• The data $f$ lives in the dual $V^* = H^{-1}$.

• The bilinear form $B : V \times V \to \mathbb{R}$ connects them.

• The pivot $H = L^2$ sits in between, providing the $L^2$ inner product that mediates between function values and functionals.

What Sobolev spaces provide¶

Compared to the fundamental solution approach, the weak formulation gains:

1. Boundary conditions. The space $H^1_0(\Omega)$ encodes $u = 0$ on $\partial\Omega$. No need to modify the fundamental solution.

2. Variable coefficients. Replace $\int \nabla u \cdot \nabla v$ by $\int A(x)\nabla u \cdot \nabla v$ for a matrix-valued coefficient $A(x)$. As long as $A$ is uniformly elliptic, the same argument works (via Lax-Milgram instead of Riesz).

3. Regularity from the Sobolev scale. The solution $u_f$ lives in $H^1_0$. If $f$ is more regular ($f \in L^2$ rather than $H^{-1}$), elliptic regularity gives $u_f \in H^2$. The Sobolev scale tracks exactly how regular the solution is.

4. Compactness. The embedding $H^1_0(\Omega) \hookrightarrow L^2(\Omega)$ is compact (Rellich-Kondrachov), so the inverse Laplacian $(-\Delta)^{-1} : L^2 \to H^1_0 \hookrightarrow L^2$ is a compact operator. This gives eigenvalues, eigenfunctions, and the spectral theory from the linear operators chapter.

Two approaches, one solution¶

On a bounded domain $\Omega$, both approaches produce the same answer. The Green’s function $G(x,y)$ is the fundamental solution modified to satisfy boundary conditions: $-\Delta_x G(x,y) = \delta(x-y)$ with $G = 0$ on $\partial\Omega$. The convolution formula becomes

and this $u$ is also the Riesz representative in $H^1_0(\Omega)$. The fundamental solution gives the explicit kernel; the Sobolev approach gives existence without needing to find it.

Nonlinear PDE via Compactness¶

The Lax-Milgram theorem handles linear equations. For nonlinear problems, we need a different strategy: approximate, extract limits, and pass to the limit in the nonlinear terms. The tools are Banach-Alaoglu (weak compactness), Rellich-Kondrachov (compact embedding), and Sobolev embedding (controlling the nonlinearity). We illustrate with a model problem that ties together the entire theory.

The model problem¶

Consider the semilinear elliptic equation

where $\Omega \subset \mathbb{R}^d$ ($d \leq 3$) is bounded with Lipschitz boundary and $f \in H^{-1}(\Omega)$. The cubic nonlinearity $u^3$ models a restoring force that grows with amplitude.

The weak formulation is: find $u \in H^1_0(\Omega)$ such that

The nonlinear term $\int u^3 v \, dx$ is well-defined when $d \leq 3$ because the Sobolev embedding $H^1_0(\Omega) \hookrightarrow L^6(\Omega)$ (for $d = 3$) gives $u^3 \in L^2$ and $u^3 v \in L^1$.

Galerkin approximation¶

Choose finite-dimensional subspaces $V_n \subset H^1_0(\Omega)$ with $\bigcup_n V_n$ dense in $H^1_0(\Omega)$ (e.g., the span of the first $n$ eigenfunctions of $-\Delta$). The Galerkin problem is: find $u_n \in V_n$ such that

This is a finite-dimensional system, so Brouwer’s fixed point theorem guarantees the existence of a Galerkin solution $u_n$.

A priori bounds¶

Test with $v = u_n$ itself:

By the Poincaré inequality (Theorem 1), $\|u_n\|_{H^1} \leq C\|\nabla u_n\|_{L^2}$, so

Dividing: $\|\nabla u_n\|_{L^2} \leq C\|f\|_{H^{-1}}$. The sequence $(u_n)$ is bounded in $H^1_0(\Omega)$, uniformly in $n$.

Extracting the limit¶

The bounded sequence $(u_n)$ in the reflexive space $H^1_0(\Omega)$ has a weakly convergent subsequence by the Banach-Alaoglu theorem (Theorem 1):

Weak convergence alone is not enough to pass to the limit in the nonlinear term $u_n^3$. The map $u \mapsto u^3$ is continuous (in the strong topology), but continuous is not the same as weakly continuous: continuity preserves strong limits, and weak convergence is in general only preserved by linear bounded operators. Nonlinear maps can destroy weak limits through oscillation. The textbook example is squaring: $u_n(x) = \sin(nx)$ on $(0, 2\pi)$ satisfies $u_n \rightharpoonup 0$ in $L^2$, but

so $u \mapsto u^2$ is not weakly continuous. Cubing fails the same way: $u_n = 1 + \sin(nx)$ has $u_n \rightharpoonup 1$ but $u_n^3 \rightharpoonup \tfrac{5}{2} \neq 1 = u^3$ (the cross term $3\sin^2(nx) \rightharpoonup 3/2$ is what leaks out). So knowing only that $u_n \rightharpoonup u$ in $H^1_0$ is not enough to conclude $u_n^3 \to u^3$ in any sense; we need strong convergence.

The fix is to upgrade weak convergence to strong convergence in a space where the nonlinearity is continuous, and this is exactly what Rellich-Kondrachov provides: applying Proposition 5 with $T$ the compact embedding $\iota : H^1_0 \hookrightarrow L^p$ (compact) for $p < 6$ (taking $d=3$, so the Sobolev critical exponent is $2^* = 6$) upgrades the weak-$H^1_0$ convergence of the Galerkin sequence, along a further subsequence, to:

In particular, $u_n \to u$ in $L^4(\Omega)$. This strong convergence gives:

(since $\|u_n^3 - u^3\|_{L^{4/3}} \leq C\|u_n - u\|_{L^4}(\|u_n\|_{L^4}^2 + \|u\|_{L^4}^2)$ by the algebraic identity $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$).

Passing to the limit¶

Fix a test direction $v \in H^1_0(\Omega)$ (not just $v \in V_m$) and read each side of the Galerkin equation as a value of a functional on $H^1_0$. For every $n$, the maps

are continuous linear functionals on $H^1_0$, i.e. elements of $H^{-1}(\Omega)$. The linear piece is continuous by Cauchy-Schwarz; the nonlinear piece is continuous because $u_n \in H^1_0 \hookrightarrow L^4$ implies $u_n^3 \in L^{4/3} \hookrightarrow H^{-1}$ (Sobolev embedding dualized). So the Galerkin identity is the statement $\Phi_n = F$ in $H^{-1}$, but restricted to test directions $v \in V_m$, i.e. tested only against a dense subset.

We want to upgrade this to $\Phi = F$ in $H^{-1}$ for a single limit object $\Phi$. Both pieces of $\Phi_n$ converge, pointwise in $v$, to the corresponding pieces built from $u$:

• Linear term. $u_n \rightharpoonup u$ in $H^1_0$ is exactly the statement $\int \nabla u_n \cdot \nabla v\,dx \to \int \nabla u \cdot \nabla v\,dx$ for every $v \in H^1_0$ (weak convergence is pointwise convergence of the Riesz functionals).

• Nonlinear term. Rellich upgraded weak-$H^1_0$ to strong-$L^4$ convergence, hence $u_n^3 \to u^3$ in $L^{4/3}$. This upgrades to convergence in $H^{-1}$ via the continuous embedding $L^{4/3} \hookrightarrow H^{-1}$: for any $g \in L^{4/3}$, Hölder and Sobolev give

  $$\Bigl|\int g\,v\,dx\Bigr| \leq \|g\|_{L^{4/3}}\,\|v\|_{L^4} \leq C\,\|g\|_{L^{4/3}}\,\|v\|_{H^1_0},$$

  (25)

  so $g$ defines a bounded linear functional on $H^1_0$ with $\|g\|_{H^{-1}} \leq C \|g\|_{L^{4/3}}$. This is exactly the dual of Sobolev $H^1_0 \hookrightarrow L^4$. Pairing with any fixed $v \in H^1_0 \subset L^4$ then gives $\int u_n^3 v\,dx \to \int u^3 v\,dx$.

So $\Phi_n(v) \to \Phi(v)$ for every $v \in H^1_0$, where $\Phi(v) = \int \nabla u \cdot \nabla v + \int u^3 v$. For each $v \in V_m$ (any $m$) the identity $\Phi_n(v) = F(v)$ holds eventually, so passing $n \to \infty$ yields $\Phi(v) = F(v)$ on the dense set $\bigcup_m V_m$. Continuity of $\Phi$ and $F$ on $H^1_0$ extends this to all $v \in H^1_0$. The Galerkin equation has upgraded to the identity $\Phi = F$ in $H^{-1}$, which is the weak form $-\Delta u + u^3 = f$ we set out to solve.

Linear vs. nonlinear: why compactness was needed¶

Compare with the purely linear problem $-\Delta u = f$ from the start of the section. There, existence followed from Riesz representation alone: weak convergence in $H^1_0$ was already enough to pass to the limit, because the only operation applied to $u_n$ was the continuous linear functional $v \mapsto \int \nabla u_n \cdot \nabla v$. Linear bounded operators preserve weak convergence by definition, so weak subsequential limits suffice and Rellich is never needed.

Nonlinearity changes this. The map $u \mapsto u^3$ is continuous but not weakly continuous, so weak convergence of $u_n$ gives no information about $u_n^3$. The fix is to upgrade weak convergence to strong convergence in a space where the nonlinearity is continuous, and this upgrade is precisely what a compact embedding provides. This is the general shape of the direct method: for linear problems weak compactness (Banach-Alaoglu, Riesz) is enough; for nonlinear problems weak compactness must be combined with a compact embedding (Rellich-Kondrachov) to tame the nonlinear term.

Why multiplication is the hard operation. The obstruction is structural, and the Banach-algebra chapter already told us what it is: multiplication is a frequency-convolution operation ($\widehat{uv} = \hat u * \hat v$), so squaring or cubing a function sums its frequencies. Two modes at frequencies $n$ and $m$ produce a mode at frequency $n+m$. High-frequency content in $u_n$ therefore breeds even higher frequencies in $u_n^3$, so $u_n^3$ can be strictly worse behaved than $u_n$, and may leave the Sobolev space that $u_n$ lived in. That is why Sobolev spaces fail to be Banach algebras below the threshold $s > d/p$ (Theorem 1): squaring moves you out of the space, because the squared spectrum escapes the regularity bound. The compactness trick sidesteps the problem by reducing the question to strong convergence in a concrete $L^q$, where multiplication is continuous (Hölder), rather than working in the $H^s$ where multiplication might blow up the regularity budget.