Given a function u, we consider its quasi–convex envelope u* and we investigate the relationship between the Hessian matrices of u and u* (the latter intended in viscosity sense); we obtain two inequalities between the tangential Laplacians of u and u* and the normal second derivatives of u and u* (the words tangential and normal are referred to a level set of the involved functions). Then we apply the result to prove convexity of level sets of solutions of elliptic equations in convex rings. Our results can be applied to a class of elliptic operator which can be naturally decomposed in a tangential and a normal part, such as Laplacian, p–Laplacian or the Mean Curvature operator.
Quasi-concave envelope of a function and convexity of level sets of solutions to elliptic equations / A. COLESANTI; P. SALANI. - In: MATHEMATISCHE NACHRICHTEN. - ISSN 0025-584X. - STAMPA. - 258:(2003), pp. 3-15.
Quasi-concave envelope of a function and convexity of level sets of solutions to elliptic equations
COLESANTI, ANDREA;SALANI, PAOLO
2003
Abstract
Given a function u, we consider its quasi–convex envelope u* and we investigate the relationship between the Hessian matrices of u and u* (the latter intended in viscosity sense); we obtain two inequalities between the tangential Laplacians of u and u* and the normal second derivatives of u and u* (the words tangential and normal are referred to a level set of the involved functions). Then we apply the result to prove convexity of level sets of solutions of elliptic equations in convex rings. Our results can be applied to a class of elliptic operator which can be naturally decomposed in a tangential and a normal part, such as Laplacian, p–Laplacian or the Mean Curvature operator.File | Dimensione | Formato | |
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