Fredholm boundary problems and solvability

Let Omega be a bounded Lipschitz domain and consider L-zero equal to minus divergence of A gradient plus c-zero, with homogeneous Dirichlet conditions. Assume A is bounded measurable, real symmetric and uniformly elliptic, and c-zero is bounded nonnegative. Lax–Milgram gives an inverse from L-two sources into H-zero-one. The embedding of H-zero-one into L-two is compact on this bounded domain, so the same inverse K, viewed as an L-two-to-L-two map, is compact.

Now subtract a real scalar sigma times the identity from L-zero. Solving the weak equation can be rewritten u minus sigma K u equals K f. This is identity minus a compact operator, which is Fredholm of index zero. Its kernel is finite-dimensional and its range is closed. In the symmetric setting K is self-adjoint, so resonance corresponds to a Dirichlet eigenvalue of L-zero. The reduction does not require pretending that an unbounded differential operator is bounded on all of L-two.

If a source is generated by Lu and v lies in the kernel of the adjoint boundary problem, the weak adjoint identity forces the source pairing with v to vanish. Fredholm's alternative makes those necessary orthogonality conditions sufficient in this compact-resolvent setting. For a self-adjoint operator the adjoint kernel is the same as the original kernel. In a nonsymmetric problem it generally is not; testing against only the original homogeneous modes can miss the correct obstruction.

If the kernel is zero, every L-two source has one unique weak solution. If the kernel is nonzero, only compatible sources have solutions, and every such solution may be modified by any kernel element. A normalization, such as orthogonality to the kernel, selects one representative after existence has been established. It cannot rescue an incompatible source. Solving a discretized linear system with an arbitrary small regularization may hide this distinction unless the residual and regularization dependence are also checked.

For the Laplacian with homogeneous Neumann conditions on a bounded connected smooth domain, constants lie in the kernel. Integrating the equation shows that a zero-flux source must have zero spatial mean. A zero-mean normalization then fixes the otherwise free additive constant. With nonzero boundary flux, the compatibility condition includes that flux with the sign dictated by minus the Laplacian. It is not the same operator as the zero-Dirichlet problem used to construct the earlier inverse.

Solvability and regularity are also different axes. The Fredholm reduction gives a weak solution or an obstruction in the selected spaces. Upgrading that solution to H-two or a classical boundary solution needs the coefficient, source, and boundary regularity assumptions developed earlier. On unbounded domains the embedding may fail to be compact and continuous spectrum can appear, so the bounded-domain alternative cannot simply be repeated. A complete report names the operator realization, kernel, compatibility test, chosen normalization, and the actual regularity achieved.