Hessian estimates for fully nonlinear equations via the large-\(M\)-inequality principle
The authors consider the fully nonlinear elliptic equation \(F(D^{2}u,x)=f(x)\) posed in an open subset \(\Omega \) of \(\mathbb{R}^{n}\) (\(n\geq 2\)). \( F:S\times \Omega \rightarrow \mathbb{R}\) is supposed to satisfy the uniform ellipticity property: \(\nu \left\Vert Y\right\Vert \leq F(X+Y,x)-F(X,x)\leq \Lambda \left\Vert Y\right\Vert \) a.e. \(x\in \Omega \) and for every \(X,Y\in S \) the space of real \(n\times n\) symmetric matrices, \(Y\) being positive semi-definite, among other hypotheses. The purpose of the paper is to prove a local \(W^{2,p}\)-estimate for viscosity solutions to this problem. The authors here use a large \(M\)-inequality proposed by \textit{E. Acerbi} and \textit{G. Mingione} in [Duke Math. J. 136, No. 2, 285--320 (2007; Zbl 1113.35105)] and they give another proof of the seminal regularity result by \textit{L.A. Caffarelli} in [Ann. Math. (2) 130, No. 1, 189--213 (1989; Zbl 0692.35017)]. They also complete this result with weighted Hessian estimates in variable exponent spaces for viscosity solutions, assuming that \(F\) is asymptotically convex. They first define the notion of \(L^{q}\)-viscosity solution (\(q>n/2\)) as a continuous function such that if \(F(D^{2}\varphi (x),x)-f(x)\geq +\epsilon \) and \(F(D^{2}\varphi (x),x)-f(x)\leq -\epsilon \) for every \(\varphi \in W_{\mathrm{loc}}^{2,q}(\Omega )\) and a.e. \(x\in \mathcal{O}\) an open subset of \(\Omega \) and \(\epsilon >0\), then \(u-\varphi \) cannot have a local minimum in \(\mathcal{O}\). The authors here assume that \(F\) is continuous in \(X\) and measurable in \(x\), and \(f\in L_{\mathrm{loc}}^{q}(\Omega )\). They introduce the recession function \(F^{\ast }\) associated with \(F\), variable exponents \(p(\cdot)\) satisfying \(n<p_{\Omega }^{-}=\inf_{x\in \Omega }p(x)\leq \sup_{x\in \Omega }p(x)=p_{\Omega }^{+}<+\infty \) and \(\left\vert p(x)-p(y)\right\vert \leq \theta (\left\vert x-y\right\vert )\) for \(x,y\in \Omega \) and where \(\theta :[0,+\infty )\rightarrow \lbrack 0,+\infty )\) is a nondecreasing and continuous function with \(\theta (0)=0\) and \(\theta (\rho )\log (1/\rho )\leq c_{p(\cdot)}\) for every \(\rho \in (0,1)\) and for some positive constant \(c_{p(\cdot)}\). They also introduce Muckenhoupt classes \(A_{p(\cdot)}\) of weights \(w\) within this variable exponents framework. The main result assumes that \(F^{\ast }(X,x)\) exists and that \(F^{\ast }(D^{2}v,x_{0})=0\) has \(C^{1,1}\) interior estimates with constant \(c_{\ast }\) for any \(x_{0}\in \Omega \). Then there exists \(\beta _{0}\) only depending on the data such that if \(f\in L_{w,\mathrm{loc}}^{p(\cdot)}(\Omega )\) and \( \sup_{B_{r}(x_{0})\Subset \Omega }\frac{1}{\left\vert B_{r}(x_{0})\right\vert }\int_{B_{r}(x_{0})}\beta _{F^{\ast }}(x,x_{0})^{n}dx\leq \beta _{0}\) any \(L^{n}\)-viscosity solution to the problem satisfies \(D^{2}u\in L_{w,\mathrm{loc}}^{p(\cdot)}(\Omega )\). Here \(\beta _{F^{\ast }}(x,x_{0})=\sup_{X\in S\setminus \{0\}}\frac{\left\vert F^{\ast }(X,x)-F^{\ast }(X,x_{0})\right\vert }{\left\Vert X\right\Vert }\). Moreover, the authors prove estimates on \(\left\Vert D^{2}u\right\Vert _{L_{w,\mathrm{loc}}^{p(.)}(B_{\rho })}\) assuming further hypotheses on \(\rho \). For the proof, the authors prove properties of the Muckenhoupt classes, especially in the context of variable exponents. They define a concave paraboloid \(P\) with opening \(m\) as \(P(x)=l_{0}+l(x)-\frac{m}{2}\left\vert x\right\vert^{2}\) where \(l\) is a linear function and a convex paraboloid of opening \(m\) if \(m\) is replaced by \(-m\) in the preceding definition. For any continuous function \(g:U\rightarrow \mathbb{R}\) where \(U\) is an open and bounded subset of \(\mathbb{R}^{n}\), they define \(\underline{\mathcal{G}} _{m}(g,U)=\{x_{0}\in U:\) there exists a concave paraboloid \(P\) with opening \( m\) such that \(P(x_{0})=g(x_{0})\) and \(P(x)\leq g(x)\) in \(U\}\). They define \( \overline{\mathcal{G}}_{m}(g,U)\) replacing concave by convex in the preceding definition. They finally define the function \(\Theta (g,U)=\sup \{ \underline{\Theta }(g,U)(x),\overline{\Theta }(g,U)(x)\}\) where \(\underline{ \Theta }(g,U)(x)=\inf \{m>0:x\in \underline{\mathcal{G}}_{m}(g,U)\}\) and \( \overline{\Theta }(g,U)(x)=\inf \{m>0:x\in \overline{\mathcal{G}}_{m}(g,U)\}\). They establish properties of \(\Theta \). The main part of the proof consists to prove estimates with scaling on \(\Theta \). The authors conclude their paper proving the result of Caffarelli with their approach.
scientific article; zbMATH DE number 7358799
scientific article