The strong maximum principle revisited

AbstractIn this paper we first present the classical maximum principle due to E. Hopf, together with an extended commentary and discussion of Hopf's paper. We emphasize the comparison technique invented by Hopf to prove this principle, which has since become a main mathematical tool for the study of second order elliptic partial differential equations and has generated an enormous number of important applications. While Hopf's principle is generally understood to apply to linear equations, it is in fact also crucial in nonlinear theories, such as those under consideration here.In particular, we shall treat and discuss recent generalizations of the strong maximum principle, and also the compact support principle, for the case of singular quasilinear elliptic differential inequalities, under generally weak assumptions on the quasilinear operators and the nonlinearities involved. Our principal interest is in necessary and sufficient conditions for the validity of both principles; in exposing and simplifying earlier proofs of corresponding results; and in extending the conclusions to wider classes of singular operators than previously considered.The results have unexpected ramifications for other problems, as will develop from the exposition, e.g. (i)two point boundary value problems for singular quasilinear ordinary differential equations (Sections 3 and 4);(ii)the exterior Dirichlet boundary value problem (Section 5);(iii)the existence of dead cores and compact support solutions, i.e. dead cores at infinity (Section 7);(iv)Euler–Lagrange inequalities on a Riemannian manifold (Section 9);(v)comparison and uniqueness theorems for solutions of singular quasilinear differential inequalities (Section 10). The case of p-regular elliptic inequalities is briefly considered in Section 11

A comparison principle for vector valued minimizers of semilinear elliptic energy, with application to dead cores

We establish a comparison principle providing accurate upper bounds for the modulus of vector valued minimizers of an energy functional, associated when the potential is smooth, to elliptic gradient systems. Our assumptions are very mild: we assume that the potential is lower semicontinuous, and satisfies a monotonicity condition in a neighbourhood of its minimum. As a consequence, we give a sufficient condition for the existence of dead core regions, where the minimizer is equal to one of the minima of the potential. Our results extend and provide variational versions of several classical theorems, well-known for solutions of scalar semilinear elliptic PDE.Basque Government through the BERC 2018-2021 program Spanish State Research Agency through BCAM Severo Ochoa excellence accreditation SEV-2017-0718 and through project PID2020-114189RB-I00 funded by Agencia Estatal de Investigación (PID2020-114189RB-I00 / AEI / 10.13039/501100011033

The maximum principle with lack of monotonicity

We establish a maximum principle for the weighted (p, q)-Laplacian, which extends the general Pucci–Serrin strong maximum principle to this quasilinear abstract setting. The feature of our main result is that it does not require any monotonicity assumption on the nonlinearity. The proof combines a local analysis with techniques on nonlinear differential equations

The maximum principle with lack of monotonicity

We establish a maximum principle for the weighted (p, q)-Laplacian, which extends the general Pucci–Serrin strong maximum principle to this quasilinear abstract setting. The feature of our main result is that it does not require any monotonicity assumption on the nonlinearity. The proof combines a local analysis with techniques on nonlinear differential equations

The maximum principle with lack of monotonicity

We establish a maximum principle for the weighted (p, q)-Laplacian, which extends the general Pucci–Serrin strong maximum principle to this quasilinear abstract setting. The feature of our main result is that it does not require any monotonicity assumption on the nonlinearity. The proof combines a local analysis with techniques on nonlinear differential equations

Entire solutions of singular elliptic inequalities on complete manifolds

We present some qualitative properties for solutions of singular quasilinear elliptic differential inequalities on complete Riemannian manifolds, such as the validity of the weak maximum principle at infinity, and non-existence results

Existence of solutions for a class of degenerate quasilinear elliptic equation in RN with vanishing potentials

We establish the existence of positive solution for the following class of degenerate quasilinear elliptic problem(P) {-Lu-ap + V(x)vertical bar x vertical bar(-ap*)vertical bar u vertical bar(p-2)u = f(u) in R-N, u > 0 in R-N; u is an element of D-a(1,p) (R-N)where -Lu-ap = -div(vertical bar x vertical bar(-ap)vertical bar del vertical bar(p-2)del u), 1 < N, -infinity < a < N-p/p, a <= e <= a + 1, d = 1 + a - e, and p* = p* (a, e) = Np/N-dp denote the Hardy-Sobolev's critical exponent, V is a bounded nonnegative vanishing potential and f has a subcritical growth at infinity. The technique used here is a truncation argument together with the variational approach

Entire Minimizers of Allen–Cahn Systems with Sub-Quadratic Potentials

We study entire minimizers of the Allen–Cahn systems. The specific feature of our systems are potentials having a finite number of global minima, with sub-quadratic behaviour locally near their minima. The corresponding formal Euler–Lagrange equations are supplemented with free boundaries. We do not study regularity issues but focus on qualitative aspects. We show the existence of entire solutions in an equivariant setting connecting the minima of W at infinity, thus modeling many coexisting phases, possessing free boundaries and minimizing energy in the symmetry class. We also present a very modest result of existence of free boundaries under no symmetry hypotheses. The existence of a free boundary can be related to the existence of a specific sub-quadratic feature, a dead core, whose size is also quantified

Étude de quelques problèmes elliptiques et paraboliques quasi-linéaires avec singularités

Cette thèse s inscrit dans le domaine mathématique de l analyse des équations aux dérivées partielles non-linéaires. Plus précisément, nous avons fait ici l étude de problèmes quasi-linéaires singuliers. Le terme "singulier" fait référence à l intervention d une non-linéarité qui explose au bord du domaine où équation est posée. La présence d une telle singularité entraîne un manque de régularité et donc de compacité des solutions qui ne nous permet pas d appliquer directement les méthodes classiques de l analyse non-linéaire pour démontrer l existence de solutions et discuter des propriétés de régularité et de comportement asymptotique de ces solutions. Pour contourner cette difficulté, nous sommes amenés à établir des estimations a priori très fines au voisinage du bord du domaine en combinant diverses méthodes : méthodes de monotonie (reliée au principe du maximum), méthodes variationnelles, argument de convexité, méthodes de point fixe et semi-discrétisation en temps. A travers, l étude de trois problèmes-modèle faisant intervenir l opérateur p-Laplacien, nous avons montré comment ces différentes méthodes pouvaient être mises en œuvre. Les résultats que nous avons obtenus sont décrits dans les trois chapitres de cette thèse : Dans le Chapitre I, nous avons étudié un problème d absorption elliptique singulier. En utilisant des méthodes de sur- et sous solutions et des méthodes variationnelles, nous établissons des résultats d existence de solutions. Par des méthodes de comparaison locale, nous démontrons également la propriété de support compact de ces solutions, pour de fortes singularités. Dans le Chapitre II, nous étudions le cas d un système d équations quasi-linéaires singulières. Par des arguments de point fixe et de monotonie, nous démontrons deux résultats généraux d existence de solutions. Dans un deuxième temps, nous faisons une analyse plus détaillée de systèmes du type Gierer-Meinhardt modélisant des phénomènes biologiques. Des résultats d unicité ainsi que des estimations précises sur le comportement des solutions sont alors obtenus. Dans le Chapitre III, nous faisons l étude d un problème d absorption, parabolique singulier. Nous établissons par une méthode de semi-discrétisation en temps des résultats d existence de solutions. Grâce à des inégalités d énergie, nous démontrons également l extinction en temps fini de ces solutions.This thesis deals with the mathematical field of nonlinear partial differential equations analysis. More precisely, we focus on quasilinear and singular problems. By singularity, we mean that the problems that we have considered involve a nonlinearity in the equation which blows-up near the boundary. This singular pattern gives rise to a lack of regularity and compactness that prevent the straightforward applications of classical methods in nonlinear analysis used for proving existence of solutions and for establishing the regularity properties and the asymptotic behavior of the solutions. To overcome this difficulty, we establish estimations on the precise behavior of the solutions near the boundary combining several techniques : monotonicity method (related to the maximum principle), variational method, convexity arguments, fixed point methods and semi-discretization in time. Throughout the study of three problems involving the p-Laplacian operator, we show how to apply this different methods. The three chapters of this dissertation the describes results we get : In Chapter I, we study a singular elliptic absorption problem. By using sub- and super-solutions and variational methods, we prove the existence of the solutions. In the case of a strong singularity, by using local comparison techniques, we also prove that the compact support of the solution. In Chapter II, we study a singular elliptic system. By using fixed point and monotonicity arguments, we establish two general theorems on the existence of solution. In a second time, we more precisely analyse the Gierer-Meinhardt systems which model some biological phenomena. We prove some results about the uniqueness and the precise behavior of the solutions. In Chapter III, we study a singular parabolic absorption problem. By using a semi-discretization in time method, we establish the existence of a solution. Moreover, by using differential energy inequalities, we prove that the solution vanishes in finite time. This phenomenon is called "quenching".PAU-BU Sciences (644452103) / SudocSudocFranceF