Author:
Monticelli D. D.,Punzo F.
Abstract
AbstractWe obtain weighted Poincaré inequalities in bounded domains, where the weight is given by a symmetric nonnegative definite matrix, which can degenerate on submanifolds. Furthermore, we investigate uniqueness and nonuniqueness of solutions to degenerate elliptic and parabolic problems, where the diffusion matrix can degenerate on subsets of the boundary of the domain. Both the results are obtained by means of the distance function from the degeneracy set, which is used to construct suitable local sub– and supersolutions.
Funder
Istituto Nazionale di Alta Matematica "Francesco Severi"
Ministero dell’Università e della Ricerca
Publisher
Springer Science and Business Media LLC
Reference13 articles.
1. Cianchi, A., Maz’ya, V.G.: G. Global boundedness of the gradient for a class of nonlinear elliptic systems. Arch. Ration. Mech. Anal. 212(1), 129–177 (2014)
2. Fichera, G. On a unified theory of boundary value problems for elliptic-parabolic equations of second order, In: “Boundary Problems in Differential Equations”, pp. 97-120, Univ. of Wisconsin Press, Madison, (1960)
3. Foote, R.L.: Regularity of the distance function. Proc. AMS 92, 153–155 (1984)
4. A. Friedman, Stochastic Differential Equations and Applications, I, II (Academic Press, 1976)
5. Kranz, S., Parks, H.: Distance to $${C}^{k}$$ hypersurfaces. J. Diff. Eq. 40, 116–120 (1981)