12.3 Weak solutions
Definition 12.3.1. Let \(f\) be square-integrable. A weak solution of \(-\Delta u = f\) in \(\Omega \), \(u = 0\) on \(\partial \Omega \), is a \(u \in H^1_0(\Omega )\) with \[ \int _\Omega \nabla u\cdot \nabla v\,dx = \int _\Omega fv\,dx \qquad \text {for all } v \in H^1_0(\Omega ) . \]
A classical solution is a weak solution, by integration by parts. Non-zero boundary values \(g\) are handled by writing \(u = u_g + w\) with \(u_g\) any extension of \(g\) and \(w\) solving a problem with zero boundary values.
Theorem 12.3.2 (Existence of weak solutions). For every square-integrable \(f\) there is exactly one weak solution \(u\), it minimises \(\frac 12\|w\|_E^2 - \int fw\) over \(H^1_0(\Omega )\), and \(\|u\|_E \leq (d/\pi )\|f\|_{L^2}\).
Proof. The functional \(\ell (v) = \int fv\) is linear, and by Cauchy–Schwarz and Poincaré, \(|\ell (v)| \leq \|f\|_{L^2}\|v\|_{L^2} \leq (d/\pi )\|f\|_{L^2}\|v\|_E\), so it is bounded on the Hilbert space \(H^1_0\). By the Riesz representation theorem there is exactly one \(u \in H^1_0\) with \((u, v)_E = \ell (v)\) for all \(v\), which is the definition of a weak solution, and \(\|u\|_E = \|\ell \| \leq (d/\pi )\|f\|_{L^2}\). The minimising property is the computation in the proof of Theorem 12.1.1, which used only the weak form. □
For an equation \(-\operatorname {div}(A(x)\nabla u) + b\cdot \nabla u + cu = f\) the bilinear form \(a(u, v) = \int (A\nabla u\cdot \nabla v + (b\cdot \nabla u)v + cuv)\) need not be symmetric, and the Riesz theorem is replaced by the Lax–Milgram theorem: if \(a\) is bounded and coercive, \(a(v, v) \geq \alpha \|v\|_E^2\) with \(\alpha > 0\), then \(a(u, v) = \ell (v)\) for all \(v\) has exactly one solution, with \(\|u\|_E \leq \|\ell \|/\alpha \). Coercivity holds, for example, when \(A\) is uniformly positive definite, \(b = 0\) and \(c \geq 0\).
A weak solution is a solution in a generalised sense only, but the regularity theory of elliptic equations shows that it is as smooth as the data allow: if \(f\), the coefficients and the boundary are smooth, the weak solution is smooth and is a classical solution. Existence is thus proved in two steps, a soft one (Hilbert space) and a hard one (regularity), and this is the pattern of the modern theory of partial differential equations.
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