Second-order boundary estimates for solutions to a class of quasilinear elliptic equations

Small perturbations in the type of boundary conditions for an elliptic operator

The dynamics of an eco-epidemiological prey–predator model with infectious diseases in prey

The nonlinear Schrödinger equation in the finite line

This article deals with the optimization of the shape of the regions assigned to different types of boundary conditions in the definition of a ‘physical’ partial differential equation. At first, we analyze a model situation involving the solution $$u_\varOmega $$ to a Laplace equation in a domain $$\varOmega $$ ; the boundary $$\partial \varOmega $$ is divided into three parts $$\varGamma _D$$ , $$\varGamma $$ and $$\varGamma _N$$ , supporting respectively homogeneous Dirichlet, homogeneous Neumann and inhomogeneous Neumann boundary conditions. The shape derivative $$J^\prime (\varOmega )(\theta )$$ of a general objective function $$J(\varOmega )$$ of the domain is calculated in the framework of Hadamard’s method when the considered deformations $$\theta $$ are allowed to modify the geometry of $$\varGamma _D$$ , $$\varGamma $$ and $$\varGamma _N$$ (i.e. $$\theta $$ does not vanish on the boundary of these regions). The structure of this shape derivative turns out to depend very much on the regularity of $$u_\varOmega $$ near the boundaries of the regions $$\varGamma _D$$ , $$\varGamma $$ and $$\varGamma _N$$ . For this reason, in particular, $$J^\prime (\varOmega )$$ is difficult to calculate and to evaluate numerically when the transition $$\overline{\varGamma _D} \cap {\overline{\varGamma }}$$ between homogeneous Dirichlet and homogeneous Neumann boundary conditions is subject to optimization. To overcome this difficulty, an approximation method is proposed, in which the considered ‘exact’ Laplace equation with mixed boundary conditions is replaced with a ‘smoothed’ version, featuring Robin boundary conditions on the whole boundary $$\partial \varOmega $$ with coefficients depending on a small parameter $$\varepsilon $$ . We prove the consistency of this approach in our model context: the approximate objective function $$J_\varepsilon (\varOmega )$$ and its shape derivative converge to their exact counterparts as $$\varepsilon $$ vanishes. Although it is rigorously justified only in a model problem, this approximation methodology may be adapted to many more complex situations, for example in three space dimensions, or in the context of the linearized elasticity system. Various numerical examples are eventually presented in order to appraise the efficiency of the proposed approximation process.

We prove theorems on the existence and regularization of periodic solutions of the wave equation with variable coefficients on an interval with homogeneous Dirichlet and Neumann boundary conditions. The nonlinear term has a power-law growth or satisfies the nonresonance condition at infinity.

We study the large time behavior of the solution of a homogenous string equation with a homogenous Dirichlet boundary condition at the left end and a homogenous Dirichlet or Neumann boundary condition at the right end. A pointwise interior actuator gives a linear viscous damping term.

Generic bifurcation of steady-state solutions

A characteristic based volume penalization method for general evolution problems applied to compressible viscous flows

Non-degenerate solutions of boundary-value problems

A general filter regularization method to solve the three dimensional Cauchy problem for inhomogeneous Helmholtz-type equations: Theory and numerical simulation

Lipschitz regularity for a general class of quasilinear elliptic equations in convex domains

This work is concerned with the periodic solution of a doubly degenerate Allen-Cahn equation with nonlocal terms associated with Neumann boundary conditions. Firstly, we define a new associated auxiliary problem. Secondly, the topological degree theorem is applied to prove the existence of a limit point to the auxiliary problem, where this limit point represents a nontrivial nonnegative time-periodic solution of the main studied problem. It is observed that the topological degree theorem technique plays an important role in proving the desired results. Furthermore, this technique can be applied to other similar equations with homogeneous Dirichlet or Neumann boundary conditions.

This work is concerned with the periodic solution of a p-Laplacian Allen-Cahn equation with nonlocal terms associated with Neumann boundary conditions. Namely, the topological degree theorem is applied to prove the existence of a limit point to the auxiliary problem, which is also considered a nontrivial nonnegative time-periodic solution of the main studied problem. The results of this work can be extended to other similar equations with homogeneous Dirichlet or Neumann boundary conditions.

A reaction–diffusion predator–prey system with homogeneous Dirichlet boundary conditions describes the lethal risk of predator and prey species on the boundary. The spatial pattern formations with the homogeneous Dirichlet boundary conditions are characterized by the Turing type linear instability of homogeneous state and bifurcation theory. Compared with homogeneous Neumann boundary conditions, we see that the homogeneous Dirichlet boundary conditions may depress the spatial patterns produced through the diffusion-induced instability. In addition, the existence of semi-trivial steady states and the global stability of the trivial steady state are characterized by the comparison technique.

We present a finite element algorithm that computes eigenvalues and eigenfunctions of the Laplace operator for two-dimensional problems with homogeneous Neumann or Dirichlet boundary conditions, or combinations of either for different parts of the boundary. We use an inverse power plus Gauss-Seidel algorithm to solve the generalized eigenvalue problem. For Neumann boundary conditions the method is much more efficient than the equivalent finite difference algorithm. We checked the algorithm by comparing the cumulative level density of the spectrum obtained numerically with the theoretical prediction given by the Weyl formula. We found a systematic deviation due to the discretization, not to the algorithm itself.