Elliptic regularity

Elliptic regularity says that solutions of an elliptic equation gain derivatives wherever the coefficients and forcing are regular. For the Laplacian, local estimates typically control two derivatives of $u$ by $\Delta u$ together with a lower-order norm of $u$.

If $u\in H^1(V)$ and $\Delta u=f\in L^2(V)$ weakly, then $u\in H_{\mathrm{loc}}^2(V)$. Test the weak equation with $-{\delta }_{-h}({\eta }^2{\delta }_{h}u)$, move the first difference quotient onto $Du$, and bound the remaining quotient on the forcing side by the first derivative of ${\eta }^2{\delta }_{h}u$. Young inequality absorbs the localized second derivatives. Uniform bounds in $h$, together with the Sobolev characterization by bounded difference quotients, give the assertion without differentiating $f$. The constant depends only on dimension and the positive distance from $U$ to the boundary of $V$.

// Target: geometry-and-topology.bigb

Once an $L^2$ distributional solution of $\Delta u=\mathrm{div}(bu)+cu+f$ gains $H^1$ regularity, expand the divergence term. Smooth multiplication makes the right side belong to $H_{\mathrm{loc}}^{m-1}$ whenever $u\in H_{\mathrm{loc}}^m$. Interior Poisson regularity gives $H_{\mathrm{loc}}^{m+1}$. Iteration and the Sobolev embedding theorem produce a smooth representative.

For $\Delta v=\mathrm{div}F+G$ and $U\subset \subset V$, a cutoff function energy estimate gives $\Vert v{\Vert }_{H^1(U)}\le C(\Vert v{\Vert }_{L^2(V)}+\Vert F{\Vert }_{L^2(V)}+\Vert G{\Vert }_{L^2(V)})$. One derivative of forcing is permitted because integration by parts places it on the test function. This estimate allows a mollified distributional solution initially in $L^2$ to gain its first weak derivative.

For a smooth uniformly elliptic homogeneous equation away from other boundaries, an interior gradient estimate on a ball of radius comparable to the distance $s$ from a flat side gives $|\nabla u|\le C{s}^{-1}{\sup }_{B_{cs}}|u|$. If $u$ extends continuously with value zero on that side, $s|\nabla u|\to 0$ locally along it. This justifies vanishing boundary terms such as $u_x\sin (nx)$ in a distributional Fourier sine projection.

For a classical solution of $\Delta u=f$ on a half-ball with zero data on its flat face, a tangential Caccioppoli inequality controls every second partial derivative having at least one tangential index. The equation recovers the remaining second normal derivative. The resulting Sobolev norm estimate has a dimension-only constant for the fixed radii one and two. Nonzero data are handled by homogenization of Dirichlet boundary data, adding the $W^{2,2}$ norm of the supplied boundary extension. No condition is imposed on the curved face because the smooth cutoff function vanishes there.

For a constant-coefficient elliptic differential operator $P(D)$ of order $N\ge 1$, localizing the equation gives $P(D)(\chi u)=\chi P(D)u+[P(D),\chi ]u$. The commutator has order at most $N-1$, so a global Fourier transform estimate raises the known Local Sobolev space index by one until the forcing index plus $N$ is reached.

Let $(w_m)$ be the orthonormal basis of Dirichlet eigenfunctions for a strictly positive Dirichlet realization of an elliptic operator, with eigenvalues ${\lambda }_m$. ThenFor integers $k$, these domains are the Sobolev domains of powers of an elliptic Dirichlet operator. At $k=0$ the condition is just Parseval identity and imposes no boundary condition. At $k=1,2$ it describes $H_0^1$ and $H^2\cap H_0^1$, respectively. Higher $k$ impose traces of powers of $L$, not only the trace of $u$. For $L=-d^2/d{x}^2$ on $(0,\pi )$, the smooth Dirichlet function $u=x(\pi -x)$ has sine coefficients proportional to $m^{-3}$ for odd $m$, so the weighted sum diverges at $k=3$ despite its ordinary $H^3$ regularity.

For a strictly positive Dirichlet realization of an elliptic operator with smooth coefficients and smooth boundary, put $X_0=L^2(U)$. For each integer $k\ge 1$,These are the domains $D(A_D^{k/2})$, with norms equivalent to the indicated Sobolev space norms. The bilinear forms $((u,v){)}_{2l}=(L^{l}u,L^{l}v{)}_{L^2}$ and $((u,v){)}_{2l+1}=B[L^{l}u,L^{l}v]$ are inner products on $X_{2l}$ and $X_{2l+1}$. The higher boundary conditions are essential: for $L=-d^2/d{x}^2$ and $u=x(\pi -x)$, $u$ is smooth and zero at the boundary but $L^{2}u=0$, so $\Vert L^{2}u{\Vert }_2$ is not a norm on all of $H^4\cap H_0^1$. Repeated elliptic regularity proves $A_D:X_{k+2}\to X_k$ is an isomorphism, which gives the norm equivalence inductively from $X_0$ and $X_1=H_0^1$.

On a smooth bounded domain, a solution offor an invertible shift satisfiesInterior regularity and boundary flattening give the derivative gain, while invertibility controls the lower-order norm.

If $U\subset {\mathbb{R}}^n$ is bounded with $C^2$ boundary, then every $F\in L^2(U)$ determines a unique $v\in H^2(U)\cap H_0^1(U)$ satisfying