4,803 research outputs found
Existence result for differential inclusion with p(x)-Laplacian
In this paper we study the nonlinear elliptic problem with p(x)-Laplacian
(hemivariational inequality). We prove the existence of a nontrivial solution.
Our approach is based on critical point theory for locally Lipschitz
functionals due to Chang
Nonlinear second-order multivalued boundary value problems
In this paper we study nonlinear second-order differential inclusions
involving the ordinary vector -Laplacian, a multivalued maximal monotone
operator and nonlinear multivalued boundary conditions. Our framework is
general and unifying and incorporates gradient systems, evolutionary
variational inequalities and the classical boundary value problems, namely the
Dirichlet, the Neumann and the periodic problems. Using notions and techniques
from the nonlinear operator theory and from multivalued analysis, we obtain
solutions for both the `convex' and `nonconvex' problems. Finally, we present
the cases of special interest, which fit into our framework, illustrating the
generality of our results.Comment: 26 page
Second-order -regularity in nonlinear elliptic problems
A second-order regularity theory is developed for solutions to a class of
quasilinear elliptic equations in divergence form, including the -Laplace
equation, with merely square-integrable right-hand side. Our results amount to
the existence and square integrability of the weak derivatives of the nonlinear
expression of the gradient under the divergence operator. This provides a
nonlinear counterpart of the classical -coercivity theory for linear
problems, which is missing in the existing literature. Both local and global
estimates are established. The latter apply to solutions to either Dirichlet or
Neumann boundary value problems. Minimal regularity on the boundary of the
domain is required. If the domain is convex, no regularity of its boundary is
needed at all
Well-posedness and regularity for a generalized fractional Cahn-Hilliard system
In this paper, we investigate a rather general system of two operator
equations that has the structure of a viscous or nonviscous Cahn--Hilliard
system in which nonlinearities of double-well type occur. Standard cases like
regular or logarithmic potentials, as well as non-differentiable potentials
involving indicator functions, are admitted. The operators appearing in the
system equations are fractional versions of general linear operators and
, where the latter are densely defined, unbounded, self-adjoint and monotone
in a Hilbert space of functions defined in a smooth domain and have compact
resolvents. We remark that our definition of the fractional power of operators
uses the approach via spectral theory. Typical cases are given by standard
second-order elliptic operators (e.g., the Laplacian) with zero Dirichlet or
Neumann boundary conditions, but also other cases like fourth-order systems or
systems involving the Stokes operator are covered by the theory. We derive
general well-posedness and regularity results that extend corresponding results
which are known for either the non-fractional Laplacian with zero Neumann
boundary condition or the fractional Laplacian with zero Dirichlet condition.
It turns out that the first eigenvalue of plays an important
und not entirely obvious role: if is positive, then the operators
and may be completely unrelated; if, however, ,
then it must be simple and the corresponding one-dimensional eigenspace has to
consist of the constant functions and to be a subset of the domain of
definition of a certain fractional power of . We are able to show general
existence, uniqueness, and regularity results for both these cases, as well as
for both the viscous and the nonviscous system.Comment: 36 pages. Key words: fractional operators, Cahn-Hilliard systems,
well-posedness, regularity of solution
Symmetrization for fractional elliptic and parabolic equations and an isoperimetric application
We develop further the theory of symmetrization of fractional Laplacian
operators contained in recent works of two of the authors. The theory leads to
optimal estimates in the form of concentration comparison inequalities for both
elliptic and parabolic equations. In this paper we extend the theory for the
so-called \emph{restricted} fractional Laplacian defined on a bounded domain
of with zero Dirichlet conditions outside of .
As an application, we derive an original proof of the corresponding fractional
Faber-Krahn inequality. We also provide a more classical variational proof of
the inequality.Comment: arXiv admin note: substantial text overlap with arXiv:1303.297
Existence of a nontrival solution for Dirichlet problem involving p(x)-Laplacian
In this paper we study the nonlinear Dirichlet problem involving
p(x)-Laplacian (hemivariational inequality) with nonsmooth potential. By using
nonsmooth critical point theory for locally Lipschitz functionals due to Chang
and the properties of variational Sobolev spaces, we establish conditions which
ensure the existence of solution for our problem.Comment: arXiv admin note: substantial text overlap with arXiv:1212.368
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