Lax–Milgram and weak Dirichlet problems

Let Omega be a bounded Lipschitz domain, let A be bounded measurable, real symmetric, and uniformly elliptic, and let c be a bounded nonnegative function. For homogeneous Dirichlet data choose H-zero-one, the closure of compactly supported smooth functions in H-one. Poincaré's inequality makes the gradient norm a complete equivalent norm on this space. Take the source F to be a continuous linear functional on that gradient-normed Hilbert space, commonly denoted an H-minus-one datum.

For smooth functions with zero boundary test data, integration by parts changes the equation minus divergence of A gradient u plus c u equals F into a bilinear identity. That identity remains meaningful for bounded measurable A and H-one functions because products of square-integrable gradients are integrable. Its extension to every H-zero-one test function is the definition of the weak solution. No normal flux trace or pointwise second derivative is required to write this particular homogeneous Dirichlet formulation.

Cauchy–Schwarz, the coefficient bound, and Poincaré show that the form is bounded by a constant times the product of the two gradient norms. Uniform ellipticity and c nonnegative show coercivity with constant lambda in that same norm. Lax–Milgram states that any bounded coercive bilinear form on a real Hilbert space represents every bounded linear functional by one unique solution. Symmetry is not required by the abstract theorem, although it holds in the simple PDE family used here.

Testing the solution identity with u gives an immediate stability estimate: its gradient norm is at most the dual source norm divided by lambda. This argument also proves uniqueness by applying it to the difference of two solutions with the same source. Keep the norm convention explicit, since using the full H-one norm changes the coercivity constant through Poincaré. Existence is not a consequence of integration by parts alone; completeness, boundedness, and coercivity are doing the functional-analytic work.

For nonzero Dirichlet data, first require an H-one extension G with the prescribed boundary trace. Write u=w+G, with w in H-zero-one. The new functional is F(v)−a(G,v), which is continuous, so the homogeneous theorem applies to w. A boundary function stated only pointwise without the required trace-space membership need not admit such a lift. The Lipschitz-domain assumption supports the usual trace framework and makes this distinction precise.

The resulting weak solution might have no classical second derivatives, particularly when coefficients jump or the boundary has corners. Interior regularity, boundary regularity, and classical solvability require stronger assumptions and later arguments. In the symmetric case the same weak solution uniquely minimizes the strictly convex energy one half a(v,v) minus F(v) over the homogeneous space. That variational interpretation is useful, but should not be imposed on a general nonsymmetric bilinear form merely because Lax–Milgram still applies.