Elliptic boundary value problem

An elliptic boundary value problem couples an elliptic differential equation in a domain to boundary conditions. Its weak formulation is often an equation for a bounded bilinear form on a Sobolev space.

A second-order operator with coefficient matrix $A(x)=(a^{ij}(x))$ is uniformly elliptic when some $\lambda >0$ satisfies $a^{ij}(x){\xi }_i{\xi }_j\ge \lambda |\xi {|}^2$ everywhere.

A second-order operator is strictly elliptic at a point when its symmetric principal coefficient matrix is positive definite there. Uniform ellipticity strengthens this pointwise condition by requiring one positive lower bound throughout the domain.

For a second-order elliptic operator $Lu=a^{ij}{D}_{ij}u+b^i{D}_{i}u+cu$ with $c\le 0$, the inequality $Lu\ge 0$ on a bounded domain impliesAn exponential barrier reduces the non-strict inequality to a contradiction at a positive interior maximum.

A Schauder estimate bounds a solution in a Hölder norm two derivatives stronger than the forcing term. For a uniformly elliptic Dirichlet problem with $C^{\alpha }$ coefficients,

For ${\Omega }^{\prime }⋐\Omega$ and a uniformly elliptic operator with $C^{0,\alpha }$ coefficients,The constant deteriorates as ${\Omega }^{\prime }$ approaches the boundary.

For a uniformly elliptic equation on a half-ball with $C^{0,\alpha }$ coefficients and $C^{2,\alpha }$ Dirichlet data $\phi$ on its flat face,

On a smooth bounded domain, interior and boundary Schauder estimates combine into a $C^{2,\alpha }$ estimate up to the full boundary. The constant depends on the domain regularity, coefficient norms, and ellipticity constants.

The Simon absorption lemma turns a scale-local estimate containing a sufficiently small multiple of the same quantity on a larger ball into a uniform estimate. Its boundary form uses balls intersected with a half-space and is the covering step that absorbs local boundary Hölder seminorms.

A degenerate elliptic operator has a positive semidefinite principal form that may lose positive definiteness at some points or jets. The p-Laplacian with $p>2$ degenerates where its solution has zero gradient.

The method of continuity joins an invertible operator to a target operator through a continuous family. Uniform a priori estimates make the set of invertible parameters both open and closed.

If a nonconstant solution of a uniformly elliptic inequality attains a boundary extremum at a point satisfying an interior sphere condition, its outward normal derivative has a strict sign.