arXiv subjects
Publications and source records attributed to Ricardo G. Duran.
We prove the stability in weighted $W^{1,1}$ spaces for standard finite element approximations of the Poisson equation in convex polygonal or polyhedral domains, when the weight belongs to Muckenhoupt's class $A_1$ and the family of meshes is quasi-uniform.
We show that, on convex polytopes and two or three dimensions, the finite element Stokes projection is stable on weighted spaces $\mathbf{W}^{1,p}_0(ω,Ω) \times L^p(ω,Ω)$, where the weight belongs to a certain Muckenhoupt class and the integrability index can be different from two. We show how this estimate can be applied to obtain error estimates for approximations of the solution to the Stokes problem with singular sources.
We analyze the approximation by mixed finite element methods of solutions of equations of the form $-\mbox{div\,} (a\nabla u) = g$, where the coefficient $a=a(x)$ can degenerate going to cero or infinity. First, we extend the classic error analysis to this case provided that the coefficient $a$ belongs to the Muckenhoupt class $A_2$. The analysis developed applies to general mixed finite element spaces satisfying the standard commutative diagram property, whenever some stability and interpolation error estimates are valid in weighted norms. Next, we consider in detail the case of Raviart-Thomas spaces of lowest order, obtaining optimal order error estimates for general regular elements as well as for some particular anisotropic ones which are of interest in problems with boundary layers. Finally we apply the results to a problem arising in the solution of the fractional Laplace equation.
The existence of right inverses of the divergence as an operator form $H^1_0(Ω)^n$ to $L_0^2(Ω)$ is a problem that has been widely studied because of its importance in the analysis of the classic equations of fluid dynamics. When $Ω$ is a bounded domain which is star-shaped with respect to a ball $B$, a right inverse given by an integral operator was introduced by Bogovskii, who also proved the continuity using the Calderón-Zygmund theory of singular integrals. In this paper we give an alternative elementary proof using the Fourier transform. As a consequence, we obtain estimates of the constant in the continuity in terms of the ratio between the diameters of $Ω$ and $B$. Moreover, using the relation between the existence of right inverses of the divergence with the Korn and improved Poincaré inequalities, we obtain estimates for the constants in these two inequalities.
This work deals with the system $(-Δ)^m u= a(x) v^p$, $(-Δ)^m v=b(x) u^q$ with Dirichlet boundary condition in a domain $Ω\subset\RR^n$, where $Ω$ is a ball if $n\ge 3$ or a smooth perturbation of a ball when $n=2$. We prove that, under appropriate conditions on the parameters ($a,b,p,q,m,n$), any non-negative solution $(u,v)$ of the system is bounded by a constant independent of $(u,v)$. Moreover, we prove that the conditions are sharp in the sense that, up to some border case, the relation on the parameters are also necessary. The case $m=1$ was considered by Souplet in \cite{PS}. Our paper generalize to $m\ge 1$ the results of that paper.
We prove a weighted version of the Hardy-Littlewood-Sobolev inequality for radially symmetric functions, and show that the range of admissible power weights appearing in the classical inequality due to Stein and Weiss can be improved in this particular case.
This paper deals with solutions of the divergence for domains with external cusps. It is known that the classic results in standard Sobolev spaces, which are basic in the variational analysis of the Stokes equations, are not valid for this class of domains. For some bounded domains $Ω\subset{\mathbb R}^n$ presenting power type cusps of integer dimension $m\le n-2$, we prove the existence of solutions of the equation ${div}{\bf u}=f$ in weighted Sobolev spaces, where the weights are powers of the distance to the cusp. The results obtained are optimal in the sense that the powers cannot be improved. As an application, we prove existence and uniqueness of solutions of the Stokes equations in appropriate spaces for cuspidal domains. Also, we obtain weighted Korn type inequalities for this class of domains.
We prove the existence of infinitely many radial solutions for elliptic systems in Rn with power weights. A key tool for the proof will be a weighted imbedding theorem for fractional-order Sobolev spaces, that could be of independent interest.