A note on solutions for asymptotically linear elliptic systems

In this paper, we are concerned with the elliptic system of
{ -△u＋V（x）u=g（x,v）, x∈R^N,
-△v＋V（x）v=f（x,u）, x∈R^N,
where V（x） is a continuous potential well, f, g are continuous and asymptotically linear as t→∞. The existence of a positive solution and ground state solution are established via variational methods.     

A note on functional a posteriori estimates for elliptic optimal control problems

In this work, new results on functional type a posteriori estimates for elliptic optimal control problems with control constraints are presented. More precisely, we derive new, sharp, guaranteed, and fully computable lower bounds for the cost functional in addition to the already existing upper bounds. Using both, the lower and the upper bounds, we arrive at two‐sided estimates for the cost functional. We prove that these bounds finally lead to sharp, guaranteed and fully computable upper estimates for the discretization error in the state and the control of the optimal control problem. First numerical tests are presented confirming the efficiency of the a posteriori estimates derived. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 403–424, 2017     

Boundary estimates for elliptic systems with L<Superscript>1</Superscript>–data

We obtain boundary estimates for the gradient of solutions to elliptic systems with Dirichlet or Neumann boundary conditions and L ¹–data, under some condition on the divergence of the data. Similar boundary estimates are obtained for div–curl and Hodge systems.     

A note on regular points for solutions of elliptic systems

A vector valued function u(x), solution of a quasilinear elliptic system cannot be too close to a straight line without being regular.     

A note on quasi-periodic solutions of some elliptic systems

We extend a recent method of proof of a theorem by Kolmogorov on the conservation of quasi-periodic motion in Hamiltonian systems so as to prove existence of (uncountably many) real-analytic quasi-periodic solutions for elliptic systems u=f _x (u, y), whereu y ^M u(y) ^N ,f=f(x, y) is a real-analytic periodic function and is a small parameter. Kolmogorov's theorem is obtained (in a special case) whenM=1 while the caseN=1 is (a special case of) a theorem by J. Moser on minimal foliations of codimension 1 on a torusT ^M +1. In the autonomous case,f=f(x), the above result holds for any .     

Gradient estimates for elliptic systems in non-smooth domains

We obtain the global W ^1,p , 1 < p < ∞, estimate for the weak solution of an elliptic system with discontinuous coefficients in non-smooth domains without using maximal function approach. It is assumed that the boundary of a bounded domain is well approximated by hyperplanes at every point and at every scale, and that the tensor coefficients belong to BMO space with their BMO semi-norms sufficiently small. S.-S. Byun was supported in part by KRF-2006-C00034 and L. Wang was supported in part by NSF Grant 0701392.     

Gradient estimates for a class of elliptic systems

Summary Gradient bounds are proved for solutions to a class of second order elliptic systems in divergence form. The main condition on this class is a generalization of the assumption that the system be the Euler-Lagrange system of equations for a functional depending only on the modulus of the gradient of the solution.     

Marcinkiewicz estimates for solutions of some elliptic problems with nonregular data

The method introduced by Ennio De Giorgi and Guido Stampacchia for the study of the regularity (L ^p , Marcinkiewicz or C ^0,α ) of the weak solutions of Dirichlet problems hinges on the handle of inequalities concerning the integral of on the subsets where |u(x)| is greater than k. In this framework, here we give a contribution with the study of the Marcinkiewicz regularity of the gradient of infinite energy solutions of Dirichlet problems with nonregular data. Dedicated to Juan Luis Vazquez for his 60th birthday (“El verano del Patriarca”, see [19]).     

Gradient estimates for elliptic systems with measurable coefficients in nonsmooth domains

We consider an elliptic system in divergence form with measurable coefficients in a nonsmooth bounded domain to find a minimal regularity requirement on the coefficients and a lower level of geometric assumption on the boundary of the domain for a global W ^1,p , 1 < p < ∞, regularity. It is proved that such a W ^1,p regularity is still available under the assumption that the coefficients are merely measurable in one variable and have small BMO semi-norms in the other variables while the domain can be locally approximated by a hyperplane, a so called δ-Reifenberg domain, which is beyond the Lipschitz category. This regularity easily extends to a certain Orlicz-Sobolev space.     

Global estimates and energy identities for elliptic systems with antisymmetric potentials

We derive global estimates in critical scale invariant norms for solutions of elliptic systems with antisymmetric potentials and almost holomorphic Hopf differential in two dimensions. Moreover, we obtain new energy identities in such norms for sequences of solutions of these systems. The results apply to harmonic maps into general target manifolds and surfaces with prescribed mean curvature. In particular, the results confirm a conjecture of Rivière in the two-dimensional setting.     

Semilinear elliptic systems with measure data

Annali di Matematica Pura ed Applicata (1923 -) - We study the Dirichlet problem for systems of the form $$-varDelta u^k=f^k(x,u)+mu ^k,,xin varOmega ,,k=1,ldots ,n$$ , where $$varOmega...     

A note on non-existence results for semi-linear cooperative elliptic systems via moving spheres

In this note, we extend some earlier non-existence, monotonicity and one-dimensionality results of W. Reichel and the author, by replacing the local Lipschitz continuity hypothesis on the non-linearities by a so-called boundedly uniform Lipschitz condition in the magnitude of .

     

estimates in homogenization of elliptic systems with measurable coefficients

We prove an optimal theorem for a weak solution of an elliptic system in divergence form with measurable coefficients in a homogenization problem. Our theorem is sharp with respect to the assumption on the coefficients. Indeed, we allow the very rapidly oscillating coefficients to be merely measurable in one variable.     

Gradient estimates for a class of quasilinear elliptic equations with measure data

We study a class of non-homogeneous quasilinear elliptic equations with measure data to obtain an optimal regularity estimate. We prove that the gradient of a weak solution to the problem is as integrable as the first order maximal function of the associated measure in the Orlicz spaces up to a correct power.