No abstract

We prove a new, universal gradient continuity estimate for solutions to quasilinear equations with varying coefficients at points on its critical singular set of degeneracy S(u) := {X : Du(X) = 0}. Our main Theorem reveals that along S(u), u is asymptotically as regular as solutions to constant coefficient equations. In particular, along the critical set S(u), Du enjoys a modulus of continuity much superior than the, possibly low, continuity feature of the coefficients. The results are new even in the context of linear elliptic equations, where it is herein shown that H 1 -weak solutions to div (aij (X)Du) = 0, with aij elliptic and Dini-continuous are actually C 1,1 − along S(u). The results and insights of this work foster a new understanding on smoothness properties of solutions to degenerate or singular equations, beyond typical elliptic regularity estimates, precisely where the diffusion attributes of the equation collapse.

We establish a new oscillation estimate for solutions of nonlinear partial differential equations of elliptic, degenerate type. This new tool yields a precise control on the growth rate of solutions near their set of critical points, where ellipticity degenerates. As a consequence, we are able to prove the planar counterpart of the longstanding conjecture that solutions of the degenerate p-Poisson equation with a bounded source are locally of class C p ′ = C 1, 1 p−1 ; this regularity is optimal.

Abstract. σ k -Yamabe equations are conformally invariant equations generalizing the classical Yamabe equation. In [38] YanYan Li proved that an admissible solution with an isolated singularity at 0 ∈ R n to the σ k -Yamabe equation is asymptotically radially symmetric. In this work we prove that an admissible solution with an isolated singularity at 0 ∈ R n to the σ k -Yamabe equation is asymptotic to a radial solution to the same equation on R n \ {0}. These results generalize earlier pioneering work in this direction on the classical Yamabe equation by Caffarelli, Gidas, and Spruck. In extending the work of Caffarelli et al, we formulate and prove a general asymptotic approximation result for solutions to certain ODEs which include the case for scalar curvature and σ k curvature cases. An alternative proof is also provided using analysis of the linearized operators at the radial solutions, along the lines of approach in a work by Korevaar, Mazzeo, Pacard, and Schoen.

This paper provides universal, optimal moduli of continuity for viscosity solutions to fully nonlinear elliptic equations F (X, D 2 u) = f (X), based on weakest integrability properties of f in different scenarios. The primary result established in this work is a sharp Log-Lipschitz estimate on u based on the L n norm of f , which corresponds to optimal regularity bounds for the critical threshold case. Optimal C 1,α regularity estimates are delivered when f ∈ L n+ǫ . The limiting upper borderline case, f ∈ L ∞ , also has transcendental importance to elliptic regularity theory and its applications. In this paper we show, under convexity assumption on F , that u ∈ C 1,Log−Lip , provided f has bounded mean oscillation. Once more, such an estimate is optimal. For the lower borderline integrability condition allowed by the theory, we establish interior a priori estimates on the C 0, n−2ε n−ε norm of u based on the L n−ε norm of f , where ε is the Escauriaza universal constant. The exponent n−2ε n−ε is optimal. When the source function f lies in L q , n > q > n − ε, we also obtain the exact, improved sharp Hölder exponent of continuity.MSC: 35B65, 35J60.

We study transmission problems with free interfaces from one random medium to another. Solutions are required to solve distinct partial differential equations, L + and L − , within their positive and negative sets respectively. A corresponding flux balance from one phase to another is also imposed. We establish existence and L ∞ bounds of solutions. We also prove that variational solutions are non-degenerate and develop the regularity theory for solutions of such free boundary problems.

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.

In this paper we study one-phase fully nonlinear singularly perturbed elliptic problems with high energy activation potentials, ζ ε (u) with ζ ε → δ 0 · ζ . We establish uniform and optimal gradient estimates of solutions and prove that minimal solutions are non-degenerated. For problems governed by concave equations, we establish uniform weak geometric properties of approximating level surfaces. We also provide a thorough analysis of the free boundary problem obtained as a limit as the ε-parameter term goes to zero. We find the precise jumping condition of limiting solutions through the phase transition, which involves a subtle homogenization process of the governing fully nonlinear operator. In particular, for rotational invariant operators, F (D 2 u), we show the normal derivative of limiting function is constant along the interface. Smoothness properties of the free boundary are also addressed.