Nonhomogeneous dirichlet problem for elliptic operators with axially discontinuous coefficients

Si studia, in un cilindro, il problema di Dirichlet per l'equazione ellittica del II ordine: $\text{L}{}_{\mathrm{\alpha }}\text{u = ?}$, dove $\text{L}{}_{\mathrm{\alpha }}\text{=}\text{αΔ + (1 ? 3α)∑}{}_{\mathrm{ij\; =\; 1}}{}^2\text{x}{}_{\mathrm{i}}\text{x}{}_{\mathrm{j}}\text{(x}{}_1{}^2\text{+ x}{}_2{}^2\text{)}{}^{\mathrm{?1}}\text{?}{}^2\text{?x}{}_{\mathrm{i}}\text{?x}{}_{\mathrm{j}}\text{, α ? (0,}\text{1}\text{3}\text{]}$è l'operatore a coefficienti discontinui sull'asse x₃ già introdotto da N. Ural'tseva per mostrare che l'equazione considerata può non avere soluzione nello spazio di Sobolev W^2,p(p > 2) per qualche f?L^p. In questo lavoro si danno limitazioni a priori e teoremi di esistenza e unicità in W^2,p quando p varia in un intervallo (p₁(α), p₂(α)), dipendente dalla costante di ellitticità α. Se p = p₂(α) le limitazioni a priori cadono: l'esempio è quello di Ural'tseva.     

Uniqueness in the dirichlet problem for some elliptic operators with discontinuous coefficients

Uniqueness is proved for the Dirichlet problem for second order nondivergence form elliptic operators with coefficients continuous except at a countable set of points having at most one accumulation point. Moreover, gradient estimates are proved.The authors are partially supported by the National Science Foundation Grant no. NSF/DMS 8421377-04.     

The Dirichlet problem for second order elliptic equations with coefficients in the Stummel class in unbounded domains

In this note we prowe existence and unicity of solution of a Dirichlet problem for second order elliptic operator in the divergence form, with the coefficients of the lower order terms belonging to a variant of the Stummel-Kato class, in an unbounded domain, extending the works [6] and [2].

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Sunto In questa nota proviamo un Teorema di esistenza e unicità per la soluzione di un problema di Dirichlet relativo ad un operatore ellittico del secondo ordine in forma di divergenza, con i coefficienti dei termini di ordine inferiore appartenenti ad una variante dello spazio di Stummel-Kato, in un dominio non limitato, estendendo i lavori [6] e [2].

     

Strong unique continuation for higher order elliptic equations with Gevrey coefficients

We address the strong unique continuation problem for higher order elliptic partial differential equations in 2D with Gevrey coefficients. We provide a quantitative estimate of unique continuation (observability estimate) and prove that the solutions satisfy the strong unique continuation property for ranges of the Gevrey exponent strictly including non-analytic Gevrey classes. As an application, we obtain a new upper bound on the Hausdorff length of the nodal sets of solutions with a polynomial dependence on the coefficients.     

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.     

Solutions of the dirichlet problem for elliptic systems in a disk

We study 2×2 second-order elliptic systems, which can be written as a single equation with complex coefficients. In an arbitrary bounded region with smooth boundary, we obtain necessary and sufficient conditions on the trace relation of a solution, which we apply in the case of a disk. We prove existence and uniqueness theorems for a solution in a Sobolevskii space of an equation which is not properly elliptic. In particular, we prove that the properties of the problem determine the angle between the bicharacteristics. If it is -rational, then there is no uniqueness, but if it is -irrational, then the smoothness of the solution of the Dirichlet problem depends on the order of its approximation by -rational numbers; but if it is nonreal, then the problem has the usual properties for the elliptic case.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 10, pp. 1307–1313, October, 1992.     

Higher order elliptic and parabolic systems with variably partially BMO coefficients in regular and irregular domains

The solvability in Sobolev spaces is proved for divergence form complex-valued higher order parabolic systems in the whole space, on a half-space, and on a Reifenberg flat domain. The leading coefficients are assumed to be merely measurable in one spacial direction and have small mean oscillations in the orthogonal directions on each small cylinder. The directions in which the coefficients are only measurable vary depending on each cylinder. The corresponding elliptic problem is also considered.     

Gradient estimates for higher order elliptic equations on nonsmooth domains

We establish optimal gradient estimates in Orlicz space for a nonhomogeneous elliptic equation of higher order with discontinuous coefficients on a nonsmooth domain. Our assumption is that for each point and for each sufficiently small scale the coefficients have small mean oscillation and the boundary of the domain is sufficiently close to a hyperplane. As a consequence we prove the classical Wm,p, m=1,2,…, 1<p<∞, estimates for such a higher order equation. Our results easily extend to higher order elliptic and parabolic systems.     

The Dirichlet problem of higher order quasilinear elliptic equation

The paper is concerned with the Dirichlet problem of higher order quasilinear elliptic equation:

     

The dirichlet problem for sub-elliptic second order equations

The C-regularity up to the boundary of solutions to the Dirichlet problem: is proved, using a comparison principle of L with a Hörmander's type operator X _j ^* X_j, where is a smooth bounded open subset of Rⁿ, and is a second-order degenerate elliptic operator with smooth coefficients, satisfying the so-called Fefferman-Phong's condition.     

On boundary values of solutions of the dirichlet problem for second order elliptic equation

The paper suggests some conditions on the lower order terms, which provide that the solution of the Dirichlet problem for the general elliptic equation of the second order

On theg-convergence of nonlinear elliptic operators related to the dirichlet problem in variable domains

A concept ofG-convergence of operatorsA _s:W _s W _s ^* to an operatorA:W W ^* is introduced and studied under a certain relationship between Banach spacesW _s,s=1,2, ..., and a Banach spaceW. It is shown that conditions establishing this relationship for abstract spaces are satisfied by the Sobolev spacesW ^k,m ( _s) andW ^k,m(), where { _s} is a sequence of perforated domains contained in a bounded region R ⁿ. Hence, the results obtained for abstract operators can be applied to the operators of the Dirichlet problem in the domains _s.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 7, pp. 948–962, July, 1993.