Singular Boundary Data Research Articles

On a localization-in-frequency approach for a class of elliptic problems with singular boundary data

We consider a class of nonhomogeneous elliptic equations in the half-space with critical singular boundary potentials and nonlinear fractional derivative terms. The forcing terms are considered on the boundary and can be taken as singular measure. Employing a functional setting and approach based on localization-in-frequency and Littlewood–Paley decomposition, we obtain results on solvability, regularity, and symmetry of solutions.

Singular solutions for space-time fractional equations in a bounded domain

This paper is devoted to describing a linear diffusion problem involving fractional-in-time derivatives and self-adjoint integro-differential space operators posed in bounded domains. One main concern of our paper is to deal with singular boundary data which are typical of fractional diffusion operators in space, and the other one is the consideration of the fractional-in-time Caputo and Riemann–Liouville derivatives in a unified way. We first construct classical solutions of our problems using the spectral theory and discussing the corresponding fractional-in-time ordinary differential equations. We take advantage of the duality between these fractional-in-time derivatives to introduce the notion of weak-dual solution for weighted-integrable data. As the main result of the paper, we prove the well-posedness of the initial and boundary-value problems in this sense.

Approximation of singular solutions and singular data for Maxwell’s equations by Lagrange elements

Abstract A Lagrange finite element method is proposed for Maxwell’s equations in Lipschitz domains. The method is suitable for the approximation of singular solutions lying outside $(H^1(\varOmega ))^3$, with nonhomogeneous singular boundary data in the tangential trace space of $H({\mathbf{curl}}; \varOmega )$ and a singular right-hand-side source term in $(H_0({\mathbf{curl}}; \varOmega ))^{\prime}$ (the dual space of $H_0({\mathbf{curl}}; \varOmega ))$. Numerical results are presented to illustrate performance and the theoretical results.

Blow-up phenomena in nonlocal eigenvalue problems: When theories of L1 and L2 meet

We develop a linear theory of very weak solutions for nonlocal eigenvalue problems Lu=λu+f involving integro-differential operators posed in bounded domains with homogeneous Dirichlet exterior condition, with and without singular boundary data. We consider mild hypotheses on the Green function and the standard eigenbasis of the operator. The main examples in mind are the fractional Laplacian operators.Without singular boundary datum and when λ is not an eigenvalue of the operator, we construct an L2-projected theory of solutions, which we extend to the optimal space of data for the operator L. We present a Fredholm alternative as λ tends to the eigenspace and characterise the possible blow-up limit. The main new ingredient is the transfer of orthogonality to the test function.We then extend the results to singular boundary data and study the so-called large solutions, which blow up at the boundary. For that problem we show that, for any regular value λ, there exist “large eigenfunctions” that are singular on the boundary and regular inside. We are also able to present a Fredholm alternative in this setting, as λ approaches the values of the spectrum.We also obtain a maximum principle for weighted L1 solutions when the operator is L2-positive. It yields a global blow-up phenomenon as the first eigenvalue is approached from below.Finally, we recover the classical Dirichlet problem as the fractional exponent approaches one under mild assumptions on the Green functions. Thus “large eigenfunctions” represent a purely nonlocal phenomenon.

Method of energy estimates for the study of a behavior of weak solutions of the equation of slow diffusion with singular boundary data

The equation of slow diffusion with singular boundary data is considered. An estimate of all weak solutions of such a problem is obtained, provided that the boundary regime is localized. The comparative analysis of the results obtained by the method of energy estimates and the barrier technique for the equation of porous medium is presented.

Method of energy estimates for the study of a behavior of weak solutions of the equation of slow diffusion with singular boundary data

The equation of slow diffusion with singular boundary data is considered. An estimate of all weak solutions of such a problem is obtained, provided that the boundary regime is localized. The comparative analysis of the results obtained by the method of energy estimates and the barrier technique for the equation of porous medium is presented.

Quantitative Absolute Continuity of Harmonic Measure and the Dirichlet Problem: A Survey of Recent Progress

It is a well-known folklore result that quantitative, scale invariant absolute continuity (more precisely, the weak-A∞ property) of harmonic measure with respect to surface measure, on the bound¬ary of an open set Ω ⊂ ℝn+1 with Ahlfors-David regular boundary, is equivalent to the solvability of the Dirichlet problem in Ω, with data in Lp(∂ Ω) for some p < ∞. Drawing an analogy to the famous Wiener criterion, which characterizes the domains in which the classical Dirichlet problem, with contin¬uous boundary data, can be solved, one may seek to characterize the open sets for which Lp solvability holds, thus allowing for singular boundary data. It has been known for some time that absolute continuity of harmonic measure is closely tied to rectifiability properties of ∂ Ω, but also that rectifiability alone is not sufficient to guarantee absolute continuity. In this note, we survey recent progress in this area, culminating in a geometric charac¬terization of the weak-A∞ property, and hence of solvability of the Lp Dirichlet problem for some finite p. This characterization, obtained under rather optimal background hypotheses, follows from a combination of the present author’s joint work with Martell, and the work of Azzam, Mourgoglou and Tolsa.

A sign-changing Liouville equation

We examine periodic solutions to an initial boundary value problem for a Liouville equation with sign-changing weight. A representation formula is derived for both singular and nonsingular boundary data, including data arising from fractional linear maps. In the case of singular boundary data, we study the effects that the induced singularity has on the interior regularity of solutions. Regularity criteria are also found for a generalized form of the equation.

A diffusion model in population genetics with dynamic fitness

We analyze a degenerate diffusion equation with singular boundary data, modeling the evolution of a polygenic trait under selection and drift. The equation models the contributions of a large but finite number of loci (genes) to the trait and at the same time allows the population trait mean to vary in a way that affects the strength of selection at individual loci; in this respect it differs from other population-genetic models that have been rigorously analyzed. We present existence, uniqueness and stability results for solutions of the system. We also prove that the genetic variance in the system tends to zero in the long time limit, and relate the dynamics of the trait mean to the variance.

Nonlinear diffusion and image contour enhancement

The theory of degenerate parabolic equations of the forms u_t=(\\Phi(u_x))\_{x} and v\_{t}=(\\Phi(v))\_{xx} is used to analyze the process of contour enhancement in image processing, based on the evolution model of Sethian and Malladi. The problem is studied in the framework of nonlinear diffusion equations. It turns out that the standard initial-value problem solved in this theory does not fit the present application since it it does not produce image concentration. Due to the degenerate character of the diffusivity at high gradient values, a new free boundary problem with singular boundary data can be introduced, and it can be solved by means of a non-trivial problem transformation, thus obtaining a new type of solutions that fit the desired concentration requirements. The asymptotic convergence to a sharp front is established and rates calculated.

Harmonic maps with singular boundary value from complex hyperbolic spaces into rank one symmetric spaces

In this paper we consider the Dirichlet problem at infinity of proper harmonic maps from noncompact complex hyperbolic space \(\mathbf{CH^m}\) to a rank one symmetric space N of noncompact type with singular boundary data \(f: \mathbf{S^{2m-1}}\longrightarrow \partial_\infty N\). Under some conditions on f, we show that the Dirichlet problem at infinity admits a harmonic map which assumes the boundary data f continuously.

An asymptotic analysis of the one-dimensional Vlasov–Poisson system: the Child–Langmuir law

187 Degond, P. and P.A. Raviart, An asymptotic analysis of the one-dimensional Vlasov-Poisson system: the law, Asymptotic Analysis 4 (1991) 187-214. We perform the asymptotic analysis of the one-dimensional Vlasov-Poisson system when singular boundary data are prescribed. Such a singular perturbation problem arises in the modelling of vacuum diodes under very large applied bias, and gives rise to the well-known Child-Langmuir law. In this paper, we provide a mathematical framework to this physical theory, by successively investigating the reduced problem (when the perturbation parameter e is set equal to zero) and the boundary layer problem, which gives a sharp qualitative information.

Finite element approximation of the Neumann Problem with singular boundary data

We consider in this work the finite element approximation of the Neumann Problem in a polygonal domain ofR2, in which singular boundary data are imposed.