Analyticity of solutions for semilinear elliptic systems of second order

This paper deals with systems , , where the right hand side is a -valued, real analytic function. We prove that a solution of such a system can be continued across a straight line segment , if one prescribe certain nonlinear, mixed boundary conditions on , which are assumed to be real analytic too. This continuation will be constructed by solving certain hyperbolic initial boundary value problems, generalizing an idea of H. Lewy. We apply this result to surfaces of prescribed mean curvature and to minimal surfaces in Riemannian manifolds spanned into a regular Jordan curve : Supposing analyticity of all data, we show that both types of surfaces can be continued across . Received: 29 December 2000 / Accepted: 11 July 2001 / Published online: 29 April 2002     

Optimal interior partial regularity¶for nonlinear elliptic systems: the method of A-harmonic approximation

We consider nonlinear elliptic systems of divergence type. We provide a new method for proving partial regularity for weak solutions, based on a generalization of the technique of harmonic approximation. This method is applied to both homogeneous and inhomogeneous systems, in the latter case with inhomogeneity obeying the natural growth condition. Our methods extend previous partial regularity results, directly establishing the optimal H?lder exponent for the derivative of a weak solution on its regular set. We also indicate how the technique can be applied to further simplify the proof of partial regularity for quasilinear elliptic systems. Received: 22 July 1999 / Revised version: 23 May 2000     

Concavity estimates for a class of nonlinear elliptic equations in two dimensions

We study solutions of the nonlinear elliptic equation on a bounded domain in . It is shown that the set of points where the graph of the solution has negative Gauss curvature always extends to the boundary, unless it is empty. The meethod uses an elliptic equation satisfied by an auxiliary function given by the product of the Hessian determinant and a suitable power of the solutions. As a consequence of the result, we give a new proof for power concavity of solutions to certain semilinear boundary value problems in convex domains. Received: 12 January 2000; in final form: 15 March 2001 / Published online: 4 April 2002     

Fully nonlinear elliptic equations and the dini condition

Abstract

We investigate transmission problems with strongly Lipschitz interfaces for the Dirac equation by establishing spectral estimates on an associated boundary singular integral operator, the rotation operator. Using Rellich estimates we obtain angular spectral estimates on both the essential and full spectrum for general bi-oblique transmission problems. Specializing to the normal transmission problem, we investigate transmission problems for Maxwell's equations using a nilpotent exterior/interior derivativeoperator. The fundamental commutation properties for this operator with the two basic reflection operators are proved. We show how the L ₂spectral estimates are inherited for the domain of the exterior/interior derivative operator and prove some complementary eigenvalue estimates. Finally we use a general algebraic theorem to prove a regularity property needed for Maxwell's equations.     

Some interior regularity results for solutions of Hessian equations

We prove monotonicity formulae related to degenerate k-Hessian equations which yield Morrey type estimates for certain integrands involving the second derivatives of the solution. In the special case k=2 we deduce that weak solutions in , , have locally H?lder continuous gradients. In the nondegenerate case we also show that weak solutions in , , have locally bounded second derivatives. Received February 25, 1999 / Accepted June 11, 1999 / Published online April 6, 2000     

Nonlinear elliptic equations with natural growth in general domains

We prove the existence of solutions of nonlinear elliptic equations with first-order terms having “natural growth” with respect to the gradient. The assumptions on the source terms lead to the existence of possibly unbounded solutions (though with exponential integrability). The domain Ω is allowed to have infinite Lebesgue measure. Received: April 13, 2001; in final form: September 29, 2001?Published online: July 9, 2002     

The Harnack inequality for quasilinear elliptic equations under minimal assumptions

In this note we prove the Harnack inequality for non negative solutions of the quasilinear equation

Mountain pass type solutions for quasilinear elliptic equations

We establish the existence of weak solutions in an Orlicz-Sobolev space to the Dirichlet problem where is a bounded domain in , , and the function is an increasing homeomorphism from onto . Under appropriate conditions on , , and the Orlicz-Sobolev conjugate of (conditions which reduce to subcriticality and superlinearity conditions in the case the functions are given by powers), we obtain the existence of nontrivial solutions which are of mountain pass type. Received April 22, 1999 / Accepted June 11, 1999 / Published online April 6, 2000     

Full regularity for a class of degenerated parabolic systems in two spatial variables

The authors consider quasilinear parabolic systems

Behavior of the Free Boundary Near Contact Points with the Fixed Boundary for Nonlinear Elliptic Equations

The aim of this paper is to study a free boundary problem for a uniformly elliptic fully non-linear operator. Under certain assumptions we show that free and fixed boundaries meet tangentially at contact points.     

Necessity of a Wiener type condition for boundary regularity of quasiminimizers and nonlinear elliptic equations

We use variational methods to obtain a pointwise estimate near a boundary point for quasisubminimizers of the p-energy integral and other integral functionals in doubling metric measure spaces admitting a p-Poincaré inequality. It implies a Wiener type condition necessary for boundary regularity for p-harmonic functions on metric spaces, as well as for (quasi)minimizers of various integral functionals and solutions of nonlinear elliptic equations on R ⁿ .     

Partial regularity of weak solutions for the curve shortening flow

We study the regularity of certain weak solutions for the curve shortening flow in arbitrary codimension. These solutions arise as limits of a regularization process which is related to an approach suggested by Calabi. We prove that the set of times for which such a weak solution is not smooth has Hausdorff dimension at most ?. Received: 23 May 1998 / Revised version: 7 September 1998     

Conditions for two-peaked solutions of singularly perturbed elliptic equations

We obtain necessary conditions for the existence of two-peaked solutions of singularly perturbed elliptic equations. These conditions are related to the geometry of the domain. In particular, we prove there are no two-peaked solutions in a strictly convex domain. Received: 20 January 1997 / Revised version: 2 December 1997     

Mountain pass critical points for Paneitz-Branson operators

Given a smooth compact Riemannian manifold of dimension , we study fourth order equations involving Paneitz-Branson type operators and the critical Sobolev exponent. Received: 19 June 2001 / Accepted: 27 September 2001 / Published online: 28 February 2002 Esposito's research is supported by M.U.R.S.T. under the national project Variational methods and nonlinear differential equations.     

Boundary blow-up problems for Hessian equations

This paper is concerned with large solutions of Hessian equations , in a strictly k-convex domain , i.e. solutions such that as . We prove existence, in the viscosity sense, of this kind of solutions assuming that g satisfies some natural growth conditions. Non-existence and uniqueness results are given too. Received: 18 November 1997