Comparison of solutions of nonlinear differential equations with loaded level sets

B. E. Levitsii ; A. E. Biryuk

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 6 (2016), pp. 81-98.

Résumé

We extend well-known comparison results to a class of partial differential equations with a divergent principal part containing a weight coefficient that depends on the measure of a level set of solution. Let $\Omega \subset {\mathbb{R}}^m$ be an open set with finite volume. Let $g_0(x,u) = \Phi(\mathrm{meas\left\{ \chi \in \Omega \colon u(\chi) > u(x)\right\}})$, where $\Phi$ is a continuous nonnegative function. Let $u \colon \Omega \to [0, \infty)$ be a weak solution to $$ - \sum_{j=1}^{m} \frac{\partial}{\partial x_j} \left( g(x,u) \cdot {|\nabla u|}^{p-2} \frac{\partial u}{\partial x_j}\right) = f(x) + k {|\nabla u|}^{q} $$ subject to homogeneous boundary conditions, where $g(x,u) \ge g_0(x,u), k\ge 0$ and $f \in L^1 (\Omega)$. We prove that under certain assumptions there is a weak nonnegative solution $V \colon {\Omega}^* \to [0, \infty)$ to homogeneous Dirichlet problem for $$ - \sum_{j=1}^{m} \frac{\partial}{\partial x_j} \left( g_0(x,V) \cdot {|\nabla V|}^{p-2} \frac{\partial V}{\partial x_j}\right) = f(x) + k {|\nabla V|}^{q} $$ such that $u^* \le V$ and $\int\limits_{\Omega} {|\nabla u|}^{p} dx \le \int\limits_{{\Omega}^*} {|\nabla V|}^{p} dx$. Here $\Omega^*$ is the open ball whose volume coincides with the volume of $\Omega$ and $u^*$ is the Schwarz symmetrization of $u$.

Détail

Keywords: comparison theorems, degenerate nonlinearities.
Mots-clés : $p$-elliptic equations

Bibliographie

[1] V.\;A. Zorich, Mathematical Analysis, MCCME, M., 2007, 1458 pp.

[2] A.\;N. Kolmogorov, S.\;V. Fomin, Introductory Real Analysis, Nauka Publ., M., 1989, 624 pp.

[3] I.\;P. Natanson, Theory of Functions of a Real Variable, Nauka Publ., M., 1974, 480 pp.

[4] G. Pólya, G. Szegö, Isoperimetric Inequalities in Mathematical Physics, FML Publ., M., 1962, 336 pp.

[5] G. Federer, Geometric Measure Theory, Nauka Publ., M., 1987, 760 pp.

[6] G. Hadwiger, Vorlesungen Über Inhalt, Oberfläche und Isoperimetrie, Nauka Publ., M., 1966, 416 pp.

[7] A. Alvino, S. Matarasso, G. Trombetti, “Risultati di simmetrizzazione per soluzioni di disequazioni variazionali”, Le Matematiche, 54:3 (1999), 15–28 | MR | Zbl

[8] I. Bihari, “A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations”, Acta Math. Acad. Sci. Hungar., 7:1 (1956), 81–94 | DOI | MR | Zbl

[9] K.\;M. Chong, N.\;M. Rice, Equimeasurable Rearrangements of Functions, Queen's Papers in Pure and Appl. Math, 28, Queen's University, Kingston–Ontario–Canada, 1971, 177 pp. | MR | Zbl

[10] A. Alvino, G. Trombetti, P.\;L. Lions, S. Matarasso, “Comparison results for solutions of elliptic problems via symmetrization”, Annales de l'IHP Analyse non linéaire, 16:2 (1999), 167–188 | DOI | MR | Zbl

[11] A. Alvino, G. Trombetti, J.\;I. Diaz, P.\;L. Lions, “Elliptic equations and Steiner symmetrization”, Communications on pure and applied mathematics, 49:3 (1996), 217–236 | 3.0.CO;2-G class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[12] W.\;H. Fleming, R. Rishel, “An integral formula for total gradient variation”, Archiv der Mathematik, 11:1 (1960), 218–222 | DOI | MR | Zbl

[13] S. Kesavan, Symmetrization and applications, World Scientific, New Jersey, 2006, 148 pp. | MR | Zbl

[14] C. Maderna, S. Salsa, “Dirichlet problem for elliptic equations with nonlinear first order terms: a comparison result”, Annali di Matematica pura ed applicata, 148:1 (1987), 277–288 | DOI | MR | Zbl

[15] G. Talenti, “Elliptic Equations and Rearrangements”, Ann. Scuola Norm. Sup. Pisa, Cl. Sci., IV. Ser., 3:4 (1976), 667–718 | MR