Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian
Xavier Cabré Eleonora Cinti
Discrete & Continuous Dynamical Systems - A 2010, 28(3): 1179-1206 doi: 10.3934/dcds.2010.28.1179
We establish sharp energy estimates for some solutions, such as global minimizers, monotone solutions and saddle-shaped solutions, of the fractional nonlinear equation $(-\Delta)$1/2 $u=f(u)$ in R n. Our energy estimates hold for every nonlinearity $f$ and are sharp since they are optimal for one-dimensional solutions, that is, for solutions depending only on one Euclidian variable.
   As a consequence, in dimension $n=3$, we deduce the one-dimensional symmetry of every global minimizer and of every monotone solution. This result is the analog of a conjecture of De Giorgi on one-dimensional symmetry for the classical equation $-\Delta u=f(u)$ in R n.
keywords: entire solutions. Half-Laplacian energy estimates symmetry properties
Topics in regularity and qualitative properties of solutions of nonlinear elliptic equations
Xavier Cabré
Discrete & Continuous Dynamical Systems - A 2002, 8(2): 331-359 doi: 10.3934/dcds.2002.8.331
In these notes we describe the Alexandroff-Bakelman-Pucci estimate and the Krylov-Safonov Harnack inequality for solutions of $Lu = f(x)$, where $L$ is a second order uniformly elliptic operator in nondivergence form with bounded measurable coefficients. It is the purpose of these notes to present several applications of these inequalities to the study of nonlinear elliptic equations.
The first topic is the maximum principle for the operator $L$, and its applications to the moving planes method and to symmetry properties of positive solutions of semilinear problems. The second topic is a short introduction to the regularity theory for solutions of fully nonlinear elliptic equations. We prove a $C^{1,\alpha}$ estimate for classical solutions, we introduce the notion of viscosity solution, and we study Jensen’s approximate solutions.
keywords: maximum principles Nonlinear elliptic PDE fully nonlinear equations. a priori estimates symmetry properties
Semi-stable and extremal solutions of reaction equations involving the $p$-Laplacian
Xavier Cabré Manel Sanchón
Communications on Pure & Applied Analysis 2007, 6(1): 43-67 doi: 10.3934/cpaa.2007.6.43
We consider nonnegative solutions of $-\Delta_p u=f(x,u)$, where $p>1$ and $\Delta_p$ is the $p$-Laplace operator, in a smooth bounded domain of $\mathbb R^N$ with zero Dirichlet boundary conditions. We introduce the notion of semi-stability for a solution (perhaps unbounded). We prove that certain minimizers, or one-sided minimizers, of the energy are semi-stable, and study the properties of this class of solutions.
Under some assumptions on $f$ that make its growth comparable to $u^m$, we prove that every semi-stable solution is bounded if $m < m_{c s}$. Here, $m_{c s}=m_{c s}(N,p)$ is an explicit exponent which is optimal for the boundedness of semi-stable solutions. In particular, it is bigger than the critical Sobolev exponent $p^\star-1$.
We also study a type of semi-stable solutions called extremal solutions, for which we establish optimal $L^\infty$ estimates. Moreover, we characterize singular extremal solutions by their semi-stability property when the domain is a ball and $1 < p < 2$.
keywords: 35J60 35J70; Secondary: 35J20. 35D10 Primary: 35B35
Elliptic PDE's in probability and geometry: Symmetry and regularity of solutions
Xavier Cabré
Discrete & Continuous Dynamical Systems - A 2008, 20(3): 425-457 doi: 10.3934/dcds.2008.20.425
We describe several topics within the theory of linear and nonlinear second order elliptic Partial Differential Equations. Through elementary approaches, we first explain how elliptic and parabolic PDEs are related to central issues in Probability and Geometry. This leads to several concrete equations. We classify them and describe their regularity theories. After this, most of the paper focuses on the ABP technique and its applications to the classical isoperimetric problem for which we present a new original proof, the symmetry result of Gidas-Ni-Nirenberg, and the regularity theory for fully nonlinear elliptic equations.
keywords: fully nonlinear elliptic equations. isoperimetric problem Elliptic equations probabilistic and geometric origins symmetry properties
A priori estimates for semistable solutions of semilinear elliptic equations
Xavier Cabré Manel Sanchón Joel Spruck
Discrete & Continuous Dynamical Systems - A 2016, 36(2): 601-609 doi: 10.3934/dcds.2016.36.601
We consider positive semistable solutions $u$ of $Lu+f(u)=0$ with zero Dirichlet boundary condition, where $L$ is a uniformly elliptic operator and $f\in C^2$ is a positive, nondecreasing, and convex nonlinearity which is superlinear at infinity. Under these assumptions, the boundedness of all semistable solutions is expected up to dimension $n\leq 9$, but only established for $n\leq 4$.
    In this paper we prove the $L^\infty$ bound up to dimension $n=5$ under the following further assumption on $f$: for every $\varepsilon>0$, there exist $T=T(\varepsilon)$ and $C=C(\varepsilon)$ such that $f'(t)\leq Cf(t)^{1+\varepsilon}$ for all $t>T$. This bound will follow from a $L^p$-estimate for $f'(u)$ for every $p<3$ (and for all $n\geq 2$). Under a similar but more restrictive assumption on $f$, we also prove the $L^\infty$ estimate when $n=6$. We remark that our results do not assume any lower bound on $f'$.
keywords: Semi-stable solutions regularity semilinear elliptic equations. extremal solutions boundedness

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