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### Open Access Journals

DCDS

We establish sharp energy estimates for some solutions, such as
global minimizers, monotone solutions and saddle-shaped solutions,
of the fractional nonlinear equation $(-\Delta)$

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

^{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}.
DCDS

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.

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.

CPAA

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 $1 < p < 2$.

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 $1 < p < 2$.

DCDS

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.

DCDS

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 $f'$.

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 $f'$.

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