Narrow Results

We consider the front propagation problems for the hyperbolic mean curvature motion in central force fields. Problems of this type describe the long time behavior for the electrically charged membrane with a constant charge per area and a radially symmetric potential moving with the mean curvature motion law. In this paper, we establish the existence of generalized periodic traveling waves with a large speed of propagation under appropriate assumptions on the central periodic force fields.

We prove some uniqueness results for Dirichlet problems for second-order linear elliptic partial differential equations in non-divergence form with singular data in suitable weighted Sobolev spaces, on an open subset Ω of ℝⁿ, n ≥ 2, not necessarily bounded or regular.

We analyze the behavior of solutions of nonlinear elliptic equations with nonlinear boundary conditions of type when the boundary of the domain varies very rapidly. We show that the limit boundary condition is given by , where γ(x) is a factor related to the oscillations of the boundary at point x. For the case where we have a Lipschitz deformation of the boundary, γ is a bounded function and we show the convergence of the solutions in H¹ and C^α norms and the convergence of the eigenvalues and eigenfunctions of the linearization around the solutions. If, moreover, a solution of the limit problem is hyperbolic, then we show that the perturbed equation has one and only one solution nearby.

In this paper, we are going to study the following elliptic system:

In this work we focus on the numerical approximation of the solution u of a linear elliptic PDE with stochastic coefficients. The problem is rewritten as a parametric PDE and the functional dependence of the solution on the parameters is approximated by multivariate polynomials. We first consider the stochastic Galerkin method, and rely on sharp estimates for the decay of the Fourier coefficients of the spectral expansion of u on an orthogonal polynomial basis to build a sequence of polynomial subspaces that features better convergence properties, in terms of error versus number of degrees of freedom, than standard choices such as Total Degree or Tensor Product subspaces.

We consider then the Stochastic Collocation method, and use the previous estimates to introduce a new class of Sparse Grids, based on the idea of selecting a priori the most profitable hierarchical surpluses, that, again, features better convergence properties compared to standard Smolyak or tensor product grids. Numerical results show the effectiveness of the newly introduced polynomial spaces and sparse grids.

We consider the problem -Δu = |u|^2*-2u + λu in Ω, u = 0 on ∂Ω, where Ω is an open regular subset of ℝ^N (N ≥ 3), is the critical Sobolev exponent and λ is a constant in ]0, λ₁[ where λ₁ is the first eigenvalue of -Δ. In this paper we show that, when N ≥ 4, the problem has at least (pairs of) solutions, improving a result obtained in [4] for N ≥ 6.

We prove in this work uniqueness results concerning a class of elliptic equations having gradient dependent absorption terms, whose simplest model is given in

We show that if Δu is a finite measure in Ω then, under suitable assumptions on u near ∂Ω, Δu⁺ is also a finite measure in Ω. We also study properties of the normal derivatives and on ∂Ω.

We study existence of solutions to

In this work, we consider a class of semilinear elliptic problems with nonlinear boundary conditions of mixed type. Under some monotonicity properties of the nonlinearities involved, we show that positive solutions are unique, and that their existence is characterized by the sign of some associated eigenvalues. One of the most important contributions of this work relies on the fact that we deal with boundary conditions of the form ∂u/∂ν = g(x,u) on Γ and u = 0 on Γ', where ν is the outward unit normal to Γ while Γ,Γ' are open, Γ ∩ Γ' = ∅, , but need not be disjoint.

We consider the problem

In this paper, we prove L^∞-estimates for solutions of divergence operators in the case of mixed boundary conditions. In this very general setting, the Dirichlet boundary part may be arbitrarily wild, i.e. no regularity conditions have to be imposed on it.

Multigrid algorithms, in particular, multigrid V-cycles, are investigated in this paper. We establish a general theory for convergence of the multigrid algorithm under certain approximation conditions and smoothing conditions. Our smoothing conditions are satisfied by commonly used smoothing operators including the standard Gauss–Seidel method. Our approximation conditions are verified for finite element approximation to numerical solutions of elliptic partial differential equations without any requirement of additional regularity of the solution. Our convergence analysis of multigrid algorithms can be applied to a wide range of problems. Numerical examples are also provided to illustrate the general theory.

Using deep neural networks to solve partial differential equations (PDEs) has attracted a lot of attention recently. However, why the deep learning method works is falling far behind its empirical success. In this paper, we provide a rigorous numerical analysis on the deep Ritz method (DRM) for second-order elliptic equations with Dirichlet, Neumann and Robin boundary conditions, respectively. We establish the first nonasymptotic convergence rate in $H^1$ norm for DRM using deep neural networks with smooth activation functions including logistic and hyperbolic tangent functions. Our results show how to set the hyper-parameter of depth and width to achieve the desired convergence rate in terms of the number of training samples.

Critical Sobolev-type inequality for a class of weighted Sobolev spaces on the entire space is established. We also investigate the existence of extremal function for the associated variational problem. As an application, we prove the existence of a weak solution for a general class of critical semilinear elliptic equations related to the polyharmonic operator.

A new multigrid algorithm based on the method of self-correction for the solution of elliptic problems is described. The method exploits information contained in the residual to dynamically modify the source term (right-hand side) of the elliptic problem. It has shown that the self-correcting solver is more efficient at damping the short wavelength modes of the algebraic error than its standard equivalent. When used in conjunction with a multigrid method, the resulting solver displays an improved convergence rate with no additional computational work.

In this paper, we demonstrate the application of non-commutative space-time algebra of sedeons to generalize the system of equations describing heat transfer and impurity diffusion in solids at finite velocity. It is shown that by analogy with electrodynamics, these transfer processes can be described using a compact second-order sedeonic equation for generalized scalar and vector potentials. On the one hand, this equation is reduced to the system of first-order differential equations for vortex-less mass and heat flows, and on the other hand, it can be transformed to the second-order elliptical equations for the profiles of temperature and impurity concentration. The comparison of peculiarities in transfer within the frames of parabolic and elliptic equations is discussed.

The authors consider a class of nonlinear parabolic problems where the lower order term is depending on a weighted integral of the solution, and address the issues of existence, uniqueness, stationary solutions and in some cases asymptotic behaviour.

In this paper, using an equivalent variational approach to a recent Ricceri's three critical points theorem,¹³ we obtain the existence of at least three non-trivial solutions of a Neumann problem for elliptic equations with variable exponents.