Search results for "Dirichlet problem"
showing 10 items of 79 documents
Branches of index-preserving solutions to systems of second order ODEs
2009
We investigate the existence of a continuum of index-preserving solutions to a Dirichlet problem associated with a parameter-dependent system of second order ordinary differential equations, developing a detailed analysis on the behaviour of the branches of nontrivial solutions. Our approach is based on the Rabinowitz global bifurcation Theorem combined with the notion of index and nullity of suitable linear boundary value problems. An application of the result to the study of branches of odd, periodic solutions for suitable systems of two linearly coupled pendulums of lenghts variables is also analyzed.
Existence and asymptotic properties for quasilinear elliptic equations with gradient dependence
2016
Abstract The paper focuses on a Dirichlet problem driven by the ( p , q ) -Laplacian containing a parameter μ > 0 in the principal part of the elliptic equation and a (convection) term fully depending on the solution and its gradient. Existence of solutions, uniqueness, a priori estimates, and asymptotic properties as μ → 0 and μ → ∞ are established under suitable conditions.
Generalized dirichlet problem in nonlinear potential theory
1990
The operator extending the classical solution of the Dirichlet problem for the quasilinear elliptic equation divA(x,▽u)=0 akin to thep-Laplace equation is shown to be unique providedA obeys a specific order principle. The Keldych lemma is also generalized to this nonlinear setting.
An eigenvalue Dirichlet problem involving the p-Laplacian with discontinuous nonlinearities
2005
AbstractA multiplicity result for an eigenvalue Dirichlet problem involving the p-Laplacian with discontinuous nonlinearities is obtained. The proof is based on a three critical points theorem for nondifferentiable functionals.
Regularity of renormalized solutions to nonlinear elliptic equations away from the support of measure data
2018
We prove boundedness and continuity for solutions to the Dirichlet problem for the equation $$ - {\rm{div}}(a(x,\nabla u)) = h(x,u) + \mu ,\;\;\;\;\;{\rm{in}}\;{\rm{\Omega }} \subset \mathbb{R}^{N},$$ where the left-hand side is a Leray-Lions operator from $$- {W}^{1,p}_0(\Omega)$$ into W−1,p′(Ω) with 1 < p < N, h(x,s) is a Caratheodory function which grows like ∣s∣p−1 and μ is a finite Radon measure. We prove that renormalized solutions, though not globally bounded, are Holder-continuous far from the support of μ.
Multiple solutions for a Dirichlet problem with p-Laplacian and set-valued nonlinearity
2008
AbstractThe existence of a negative solution, of a positive solution, and of a sign-changing solution to a Dirichlet eigenvalue problem with p-Laplacian and multi-valued nonlinearity is investigated via sub- and supersolution methods as well as variational techniques for nonsmooth functions.
Elliptic problems with convection terms in Orlicz spaces
2021
Abstract The existence of a solution to a Dirichlet problem, for a class of nonlinear elliptic equations, with a convection term, is established. The main novelties of the paper stand on general growth conditions on the gradient variable, and on minimal assumptions on Ω. The approach is based on the method of sub and supersolutions. The solution is a zero of an auxiliary pseudomonotone operator build via truncation techniques. We present also some examples in which we highlight the generality of our growth conditions.
Anisotropic -Laplacian equations when goes to
2010
Abstract In this paper we prove a stability result for an anisotropic elliptic problem. More precisely, we consider the Dirichlet problem for an anisotropic equation, which is as the p -Laplacian equation with respect to a group of variables and as the q -Laplacian equation with respect to the other variables ( 1 p q ), with datum f belonging to a suitable Lebesgue space. For this problem, we study the behaviour of the solutions as p goes to 1 , showing that they converge to a function u , which is almost everywhere finite, regardless of the size of the datum f . Moreover, we prove that this u is the unique solution of a limit problem having the 1-Laplacian operator with respect to the firs…
The Dirichlet problem for the total variation flow
2001
Suppose that Ω is an open bounded domain with a Lipschitz boundary. The purpose of this chapter is to study the Dirichlet problem $$ \left\{ \begin{gathered} \frac{{\partial u}} {{\partial t}} = div\left( {\frac{{Du}} {{\left| {Du} \right|}}} \right)in Q = \left( {0,\infty } \right) \times \Omega , \hfill \\ u\left( {t,x} \right) = \phi \left( x \right)on S = \left( {0,\infty } \right) \times \partial \Omega , \hfill \\ u\left( {0,x} \right) = u_0 \left( x \right)in x \in \Omega \hfill \\ \end{gathered} \right. $$ (5.1) where u0 ∈ L1(Ω) and ϕ ∈ L1 (∂Ω). This evolution equation is related to the gradient descent method used to solve the problem $$ \begin{gathered} Minimize \int {_\Omega \lef…
Optimal shape design and unilateral boundary value problems: Part II
2007
In the first part we give a general existence theorem and a regularization method for an optimal control problem where the control is a domain in R″ and where the system is governed by a state relation which includes differential equations as well as inequalities. In the second part applications for optimal shape design problems governed by the Dirichlet-Signorini boundary value problem are presented. Several numerical examples are included.