Sharp Hessian integrability estimates for nonlinear elliptic equations: An asymptotic approach

Abstract

We establish sharp W 2,p regularity estimates for viscosity solutions of fully nonlinear elliptic equations under minimal, asymptotic assumptions on the governing operator F. By means of geometric tangential methods, we show that if the recession of the operator F -formally given bywith appropriate universal estimates. Our result extends to operators with variable coefficients and in this setting they are new even under convexity of the frozen coefficient operator, M → F(x 0 , M), as oscillation is measured only at the recession level. The methods further yield BMO regularity of the hessian, provided the source lies in that space. As a final application, we establish the density of W 2,p solutions within the class of all continuous viscosity solutions, for generic fully nonlinear operators F. This result gives an alternative tool for treating common issues often faced in the theory of viscosity solutions.

A geometric tangential approach to sharp regularity for degenerate evolution equations

“…In recent years, methods from Geometric Tangential Analysis have been significantly enhanced, amplifying their range of application and providing a more user-friendly platform for advancing these endeavours (cf. [35,36,38,37,32,5,6], to cite just a few). In what follows, we shall present a small sample of problems that can be tackled by methods coming from GTA.…”

Interior Sobolev regularity for fully nonlinear parabolic equations

“…Therefore, by choosing P i+1 (x, t) := P i (x, t) + r 2iP (r −i x, r −2i t) and rescaling (21) back to the unit picture, we obtain the (i + 1) − th step of induction. Now, we proceed to the second and final part of the proof.…”

“…From a heuristic viewpoint, this operator accounts for the behavior of F at the ends of S(d), encoding an asymptotic analysis of the problem. The idea of recession function -borrowed from the realm of convex analysis -appears in the context of regularity theory in [22] and [21]. We detail the notion of recession function in Section 3.…”

A New Approach to Approximate Solutions for Nonlinear Differential Equation

“…This technique, known as the perturbation method (see [2]), has many applications in the theory of fractional differentiation operators (see [3]), in reaction-diffusion equations, stochastic stability, and asymptotic stability (see [4][5][6][7][8][9]), and for some numerical considerations (see, for example, [10][11][12]). …”