 Elliptic boundary value problem

An elliptic boundary value problem couples an elliptic differential equation in a domain to specified boundary conditions. An important example is the Poisson equationHere the Dirichlet boundary data fix the boundary value. Prescribing outward flux instead gives a Neumann boundary-value problem, which has different compatibility and normalization requirements. Thus the differential expression alone does not specify the problem.

For zero Dirichlet boundary conditions, multiply the Poisson equation by a smooth compactly supported test function and apply integration by parts:This weak formulation requires only first weak derivatives, so it makes sense for $u\in H_0^1(\Omega )$ and $f\in L^2(\Omega )$. On a bounded domain, the Poincaré inequality makes its bilinear form coercive, and the Lax-Milgram theorem gives a unique weak solution. The right side also extends to suitable forcing in the continuous dual space of $H_0^1(\Omega )$.^[ref]

The energy method explains the value of this formulation: subtracting two solutions and testing with their difference proves uniqueness. Elliptic regularity asks whether additional smoothness of coefficients, forcing and boundary upgrades the weak solution to a classical one. A Schauder estimate quantifies this gain in Hölder spaces; rough coefficients instead lead to results such as the De Giorgi-Nash-Moser theorem.