Search results for "Elliptic systems"
showing 10 items of 11 documents
Controllability-type properties for elliptic systems and applications
1991
We consider approximate and exact controllability results for elliptic problems. These results enable one to formulate optimal shape design problems in a fixed domain with certain boundary conditions.
MAPPINGS OF FINITE DISTORTION: $L^n \log^{\alpha} L$ -INTEGRABILITY
2003
Recently, systematic studies of mappings of finite distortion have emerged as a key area in geometric function theory. The connection with deformations of elastic bodies and regularity of energy minimizers in the theory of nonlinear elasticity is perhaps a primary motivation for such studies, but there are many other applications as well, particularly in holomorphic dynamics and also in the study of first order degenerate elliptic systems, for instance the Beltrami systems we consider here.
Form-perturbation theory for higher-order elliptic operators and systems by singular potentials
2020
We give a form-perturbation theory by singular potentials for scalar elliptic operators onL2(Rd)of order 2mwith Hölder continuous coefficients. The form-bounds are obtained from anL1functional analytic approach which takes advantage of both the existence ofm-gaussian kernel estimates and the holomorphy of the semigroup inL1(Rd).We also explore the (local) Kato class potentials in terms of (local) weak compactness properties. Finally, we extend the results to elliptic systems and singular matrix potentials.This article is part of the theme issue ‘Semigroup applications everywhere’.
Singular quasilinear elliptic systems involving gradient terms
2019
Abstract In this paper we establish the existence of at least one smooth positive solution for a singular quasilinear elliptic system involving gradient terms. The approach combines the sub-supersolutions method and Schauder’s fixed point theorem.
Isotropic p-harmonic systems in 2D Jacobian estimates and univalent solutions
2016
The core result of this paper is an inequality (rather tricky) for the Jacobian determinant of solutions of nonlinear elliptic systems in the plane. The model case is the isotropic (rotationally invariant) p-harmonic system ...
On critical behaviour in systems of Hamiltonian partial differential equations
2013
Abstract We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlevé-I (P $$_I$$ I ) equation or its fourth-order analogue P $$_I^2$$ I 2 . As concrete examples, we discuss nonlinear Schrödinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture.
Extremal Problems for Elliptic Systems
1998
The specific properties of optimal control problems for elliptic systems, if compared with the case of a single equation, are described. Within them are: strong closures of sets of feasible states; the relaxability via convexification; the type of necessary optimality conditions.
On necessary optimality conditions for optimal control problems governed by elliptic systems
2005
The article considers an optimal control problem for the linear elliptic system div for the case where the coefficient matrix A plays the role of control and belongs to a nonconvex set and the cost functional is a quadratic form with respect to . By transforming the original problem to a more suitable one and by using ideas from the homogenization theory a necessary optimality condition is derived.
Maximal regularity via reverse Hölder inequalities for elliptic systems of n-Laplace type involving measures
2008
In this note, we consider the regularity of solutions of the nonlinear elliptic systems of n-Laplacian type involving measures, and prove that the gradients of the solutions are in the weak Lebesgue space Ln,∞. We also obtain the a priori global and local estimates for the Ln,∞-norm of the gradients of the solutions without using BMO-estimates. The proofs are based on a new lemma on the higher integrability of functions.
Relaxation of Quasilinear Elliptic SystemsviaA-quasiconvex Envelopes
2002
We consider the weak closure WZof the set Z of all feasible pairs (solution, flow) of the family of potential elliptic systems div s0 s=1 s(x)F 0 s(ru(x )+ g(x)) f(x) =0i n; u =( u1;:::;um)2 H 1 0 (; R m ) ; =( 1;:::;s 0 )2 S; where R n is a bounded Lipschitz domain, Fs are strictly convex smooth functions with quadratic growth and S =f measurable j s(x )=0o r 1 ;s =1 ;:::;s0 ;1(x )+ +s0 (x )=1 g .W e show that WZis the zero level set for an integral functional with the integrand QF being the A-quasiconvex envelope for a certain functionF and the operator A = (curl,div) m . If the functions Fs are isotropic, then on the characteristic cone (dened by the operator A) QF coincides with the A-p…