Search results for " Boundary"
showing 10 items of 686 documents
Existence of minimizers for eigenvalues of the Dirichlet-Laplacian with a drift
2015
Abstract This paper deals with the eigenvalue problem for the operator L = − Δ − x ⋅ ∇ with Dirichlet boundary conditions. We are interested in proving the existence of a set minimizing any eigenvalue λ k of L under a suitable measure constraint suggested by the structure of the operator. More precisely we prove that for any c > 0 and k ∈ N the following minimization problem min { λ k ( Ω ) : Ω quasi-open set , ∫ Ω e | x | 2 / 2 d x ≤ c } has a solution.
Nonexistence of global weak solutions for a nonlinear Schrodinger equation in an exterior domain
2020
We study the large-time behavior of solutions to the nonlinear exterior problem L u ( t , x ) = &kappa
Fractional Laplacians in bounded domains: Killed, reflected, censored, and taboo Lévy flights.
2018
The fractional Laplacian $(- \Delta)^{\alpha /2}$, $\alpha \in (0,2)$ has many equivalent (albeit formally different) realizations as a nonlocal generator of a family of $\alpha $-stable stochastic processes in $R^n$. On the other hand, if the process is to be restricted to a bounded domain, there are many inequivalent proposals for what a boundary-data respecting fractional Laplacian should actually be. This ambiguity holds true not only for each specific choice of the process behavior at the boundary (like e.g. absorbtion, reflection, conditioning or boundary taboos), but extends as well to its particular technical implementation (Dirchlet, Neumann, etc. problems). The inferred jump-type …
Pairs of solutions for Robin problems with an indefinite and unbounded potential, resonant at zero and infinity
2018
We consider a semilinear Robin problem driven by the Laplacian plus an indefinite and unbounded potential and a Caratheodory reaction term which is resonant both at zero and $$\pm \infty $$ . Using the Lyapunov–Schmidt reduction method and critical groups (Morse theory), we show that the problem has at least two nontrivial smooth solutions.
Multiplicity of Solutions to Elliptic Problems Involving the 1-Laplacian with a Critical Gradient Term
2017
Abstract In the present paper we study the Dirichlet problem for an equation involving the 1-Laplacian and a total variation term as reaction.We prove a strong multiplicity result. Namely, we show that for any positive Radon measure concentrated in a set away from the boundary and singular with respect to a certain capacity, there exists an unbounded solution, and measures supported on disjoint sets generate different solutions.These results can be viewed as the analogue for the 1-Laplacian operator of some known multiplicity results which were first obtained by Ireneo Peral, to whom this article is dedicated, and his collaborators.
Green’s function and existence of solutions for a third-order three-point boundary value problem
2019
The solutions of third-order three-point boundary value problem x‘‘‘ + f(t, x) = 0, t ∈ [a, b], x(a) = x‘(a) = 0, x(b) = kx(η), where η ∈ (a, b), k ∈ R, f ∈ C([a, b] × R, R) and f(t, 0) ≠ 0, are the subject of this investigation. In order to establish existence and uniqueness results for the solutions, attention is focused on applications of the corresponding Green’s function. As an application, also one example is given to illustrate the result. Keywords: Green’s function, nonlinear boundary value problems, three-point boundary conditions, existence and uniqueness of solutions.
Existence of a unique solution for a third-order boundary value problem with nonlocal conditions of integral type
2021
The existence of a unique solution for a third-order boundary value problem with integral condition is proved in several ways. The main tools in the proofs are the Banach fixed point theorem and the Rus’s fixed point theorem. To compare the applicability of the obtained results, some examples are considered.
Lévy flights and Lévy-Schrödinger semigroups
2010
We analyze two different confining mechanisms for L\'{e}vy flights in the presence of external potentials. One of them is due to a conservative force in the corresponding Langevin equation. Another is implemented by Levy-Schroedinger semigroups which induce so-called topological Levy processes (Levy flights with locally modified jump rates in the master equation). Given a stationary probability function (pdf) associated with the Langevin-based fractional Fokker-Planck equation, we demonstrate that generically there exists a topological L\'{e}vy process with the very same invariant pdf and in the reverse.
Diffusion of Oxygen in Thermally Grown Oxide Scales
2009
High temperature reactivity of materials under oxidizing atmospheres is based on the formation of protective oxide scales. The protectiveness is obtained when the thermally grown oxide scales are dense, continuous and adherent to the metallic substrates (even during thermal shocks); as a matter of fact, the growth of the scale has to be governed by diffusion of species across the growing scale. The diffusing species are coming from the substrate (metallic ions) and/or from the oxidizing atmosphere (oxygen ions). The understanding of growth mechanisms can be reached by making two stage oxidation experiments, using oxygen isotopes. The experiment consists in oxidizing first the metallic subst…
Numerical Determination of Intrinsic Diffusion in Fe-Cr-Al Systems
2010
The intrinsic diffusion coefficients in diffusion aluminide coatings based on Fe-30Cr were determined at 1000oC. The diffusion fluxes were given by the Nernst Planck formulae and the Darken method for multicomponent systems was applied. This paper summarizes some numerical results to determine the composition dependent diffusivities in Fe-Cr-Al systems. The method presented in this study to obtain average intrinsic diffusion coefficients is as an alternative to the Dayananda method. Our method based on empirical parameters allowed us to predict the concentration profile during the interdiffusion process.