Sobolev gradients of viscosity supersolutions (Q6039674)

The viscosity \textit{super}solutions of the second order elliptic equation \[ F(\mathcal{H}u) =0, \qquad \mathcal{H}u=\left(\frac{\partial^2u(x)}{\partial x_i \partial x_j}\right), \] are studied in a domain \(\Omega\) in \(\mathbb{R}^n\). According to the theory of Crandall, Evans, Lions, Ishii, and others no differentiability is required in the definition for viscosity supersolutions. \textit{A priori} merely lower semicontinuity is prescribed. However, many familiar equations enjoy the property of having their viscosity supersolutions in some local Sobolev space \(W^{1,q}_{\mathrm{loc}}(\Omega)\). For example, the viscosity supersolutions of the Laplace Equation \(\Delta u=0\) (= the superharmonic functions) belong to \(W^{1,q}_{\mathrm{loc}}(\Omega)\) whenever \(0<q<n/(n-1)\). The author gives a pretty sharp characterization of the operators \(F\) having their viscosity supersolutions in some Sobolev space. The key concept is the \textit{sublevel set} \[ \Theta_{F}=\{\mathbb{X}\in \mathcal{S}^n \mid F(\mathbb{X})\leq 0\} \] in the space \(\mathcal{S}^n\) of symmetric \(n\times n\)-matrices. The \textit{asymptotic cone} \(\mathrm{ac}(\Theta_{F})\subset \mathcal{S}^n\) is essential to capture the behaviour of \(\Theta_{F}\) at infinity. The main tool to determine sharp summability exponents \(q\) is provided by the \textit{Dominative \(p\)-Laplace operator} \[ \mathcal{D}_pu\equiv\lambda_1+\lambda_2+\cdots+\lambda_n + (p-2)\lambda_n, \] where the eigenvalues \(\lambda_k(\mathcal{H}u(x))\) of the Hessian matrix are ordered: \(\lambda_1\leq \lambda_2\leq \dots\leq \lambda_n\). The Dominative \(p\)-Laplace Equation \(\mathcal{D}_pu=0\) (the Brustad Equation) happens to have the same viscosity supersolutions as the \(p\)-Laplace Equation \(\operatorname{div}(|\nabla u|^{p-2}\nabla u)=0.\) The sublevel set \(\Theta_p\) of the Dominative \(p\)-Laplace operator is of central importance. It is a convex cone in \(\mathcal{S}^n\). For the elliptic equation \(F(\mathcal{H}u)=0\) the following sufficient condition is given: \textit{Theorem 1}: Let \(2 < p \leq n,\,\,\Omega \subset \mathbb{R}^n\). If there exists an invertible matrix \(B\) such that \[ \mathrm{ac}(B^T\Theta_{F} B)\subset\Theta_p, \] then every viscosity supersolution of \(F(\mathcal{H}u)=0\) in \(\Omega\) belongs to \(W^{1,q}_{\mathrm{loc}}(\Omega)\) whenever \(0<q< n(p-1)/(n-1)\). In Theorem 2 the above condition is proved to be necessary to guarantee this \(W^{1,q}\)-property, although under natural additional assumptions: convexity and rotational invariance of the sublevel set \(\Theta_{F}\) of \(F\). There are also some results without the ellipticity assumption!