Equations In Divergence Form Research Articles

A priori bounds in [formula omitted] for solutions of elliptic equations in divergence form

We prove an a priori bound in Lp, p>1, for the solutions of the Dirichlet problem for second order linear elliptic partial differential equations in divergence form with discontinuous coefficients in unbounded domains.

On Holder continuity of solutions of second order non-uniformly degenerate parabolic equations in divergent form

On Holder continuity of solutions of second order non-uniformly degenerate parabolic equations in divergent form

A Priori Bounds in and in for Solutions of Elliptic Equations

We give an overview on some recent results concerning the study of the Dirichlet problem for second-order linear elliptic partial differential equations in divergence form and with discontinuous coefficients, in unbounded domains. The main theorem consists in an -a priori bound, . Some applications of this bound in the framework of non-variational problems, in a weighted and a non-weighted case, are also given.

Quasilinear Elliptic Equations with Morrey Data

We obtain global essential boundedness and Holder continuity of the weak solutions to quasilinear elliptic equations in divergence form with data lying in Morrey spaces.

On Removable Sets of Solutions of Neuman Problem for Quasilinear Elliptic Equations of Divergent Form

In this paper we consider a nondivergent elliptic equation of second order whose leading coefficients are from some weight space. The sufficient condition of removability of a compact with respect to this equation in the weight space of Holder functions was found.

Cauchy Problem for Semilinear Parabolic Equation with Time-Dependent Obstacles: A BSDEs Approach

In the paper we apply methods of the theory of backward stochastic differential equations to prove existence, uniqueness and stochastic representation of solutions of the Cauchy problem for semilinear parabolic equation in divergence form with two time-dependent obstacles. We consider two quite different cases: problems with distinct quasi-continuous obstacles and with irregular obstacles satisfying the so called Mokobodzki condition. As an application we also generalize the Lewy-Stampacchia inequality to non-Radon measures and give new existence result for the Dynkin game problem.

Steady subsonic potential flows through infinite multi-dimensional largely-open nozzles

We proved existence, regularity and uniqueness of steady subsonic potential flows in n-dimensional (n ≥ 3) infinite nozzles with largely-open convergent and divergent parts when the total mass flux is less than a certain value. Such nozzles consist of two cones with arbitrary open angles and an arbitrary smooth bounded tubular part. The existence of a weak solution is proved by applying the direct method of calculus of variation to a carefully chosen functional defined on a Hilbert space based upon Hardy inequality. Holder gradient regularity of weak solution is shown by using Moser iteration to quasilinear elliptic equations in divergence form. Also, the obtained solution is unique in the class of functions with finite kinetic energy by modulo a constant.

GLOBAL REGULARITY OF SOLUTIONS TO QUASILINEAR CONORMAL DERIVATIVE PROBLEM WITH CONTROLLED GROWTH

We prove the global regularity of weak solutions to a conor- mal derivative boundary value problem for quasilinear elliptic equations in divergence form on Lipschitz domains under the controlled growth con- ditions on the low order terms. The leading coefficients are inthe class of BMO functions with small mean oscillations. ( −D i (Aij(x;u)Dju + ai(x;u)) = b(x;u;r u) in ; (Aij(x;u)Dju + ai(x;u)) ��(x) = 0 on @: Here the equation is a quasilinear elliptic equation in divergence form, is a bounded Lipschitz domain in R d , d � 2, with a small Lipschitz constant, and �(x) is the outward normal vector to the surface @. We call u 2 W 1 2 () a

Local behaviour of singular solutions for nonlinear elliptic equations in divergence form

We consider the following class of nonlinear elliptic equations $$\begin{array}{ll}{-}{\rm div}(\mathcal{A}(|x|)\nabla u) +u^q=0\quad {\rm in}\; B_1(0)\setminus\{0\}, \end{array}$$ where q > 1 and \({\mathcal{A}}\) is a positive C1(0,1] function which is regularly varying at zero with index \({\vartheta}\) in (2−N,2). We prove that all isolated singularities at zero for the positive solutions are removable if and only if \({\Phi\not\in L^q(B_1(0))}\) , where \({\Phi}\) denotes the fundamental solution of \({-{\rm div}(\mathcal{A}(|x|)\nabla u)=\delta_0}\) in \({\mathcal D'(B_1(0))}\) and δ0 is the Dirac mass at 0. Moreover, we give a complete classification of the behaviour near zero of all positive solutions in the more delicate case that \({\Phi\in L^q(B_1(0))}\) . We also establish the existence of positive solutions in all the categories of such a classification. Our results apply in particular to the model case \({\mathcal{A}(|x|)=|x|^\vartheta}\) with \({\vartheta\in (2-N,2)}\) .

Harnack inequality for a class of second-order degenerate elliptic equations

A second-order degenerate elliptic equation in divergence form with a partially Muckenhoupt weight is studied. In a model case, the domain is divided by a hyperplane into two parts, and in each part the weight is a power function of |x| with the exponent less than the dimension of the space in absolute value. It is well known that solutions of such equations are Holder continuous, whereas the classical Harnack inequality is missing. In this paper, we formulate and prove the Harnack inequality corresponding to the second-order degenerate elliptic equation under consideration.

A De Giorgi–Nash type theorem for time fractional diffusion equations

We study the regularity of weak solutions to linear time fractional diffusion equations in divergence form of arbitrary time order \(\alpha \in (0,1)\). The coefficients are merely assumed to be bounded and measurable, and they satisfy a uniform parabolicity condition. Our main result is a De Giorgi–Nash type theorem, which gives an interior Holder estimate for bounded weak solutions in terms of the data and the \(L_\infty \)-bound of the solution. The proof relies on new a priori estimates for time fractional problems and uses De Giorgi’s technique and the method of non-local growth lemmas, which has been introduced recently in the context of nonlocal elliptic equations involving operators like the fractional Laplacian.

Fluctuation of Solutions to Linear Elliptic Equations with Noisy Diffusion Coefficients

We consider a linear elliptic equation in divergence form on a bounded domain (or on ℝ d ) in dimension d ≥ 2, whose coefficients are perturbed by a stationary noise of correlation length ϵ > 0. We give estimates on the fluctuation of the solution in function of the correlation length ϵ of the noise, both in terms of strong L 2 and weak L 1 norms. This result can be seen as a quantification of the propagation of uncertainties in linear elliptic partial differential equations.

Hölder continuity of solutions to parabolic equations uniformly degenerating on a part of the domain

We study a second-order parabolic equation in divergence form in a spatial domain separated in two parts by a hyperplane. The equation is uniformly parabolic in one of the parts and degenerates with respect to a small parameter $\varepsilon$ on the other part. We show that weak solutions to this equation are Hölder continuous with the Hölder exponent independent of ${\varepsilon}$.

On the Quasi-linear Elliptic PDE $${-\nabla \cdot ( \nabla{u}/ \sqrt{1-| \nabla{u} |^2}) = 4 \pi \sum_k a_k \delta_{s_k}}$$ in Physics and Geometry

It is shown that for each finite number of Dirac measures supported at points $s_n$ in three-dimensional Euclidean space, with given amplitudes $a_n$, there exists a unique real-valued Lipschitz function $u$, vanishing at infinity, which distributionally solves the quasi-linear elliptic partial differential equation of divergence form $-\nabla\cdot(\nabla{u}/\sqrt{1-|\nabla{u}|^2})=4\pi\sum_{n=1}^N a_n \delta_{s_n}$. Moreover, $u$ is real analytic away from the $s_n$. The result can be interpreted in at least two ways: (a) for any number of point charges of arbitrary magnitude and sign at prescribed locations $s_n$ in three-dimensional Euclidean space there exists a unique electrostatic field which satisfies the Maxwell-Born-Infeld field equations smoothly away from the point charges and vanishes as $|s|\to\infty$; (b) for any number of integral mean curvatures assigned to locations $s_n$ there exists a unique asymptotically flat, almost everywhere space-like maximal slice with point defects of Minkowski spacetime, having lightcone singularities over the $s_n$ but being smooth otherwise, and whose height function vanishes as $|s|\to\infty$. No struts between the point singularities ever occur.

A new proof of Wu’s theorem on vortex sheets

We give a new proof of Wu’s theorem on vortex sheets by using W1,p estimate for the elliptic equation of divergence form with partially BMO cofficients and Lp boundedness of commutators of Calderon-Zygmund operators.

On the Carleson duality

As a tool for solving the Neumann problem for divergence form equations, Kenig and Pipher introduced the space X of functions on the half space, such that the non-tangential maximal function of their L_2-Whitney averages belongs to L_2 on the boundary. In this paper, answering questions which arose from recent studies of boundary value problems by Auscher and the second author, we find the pre-dual of X, and characterize the pointwise multipliers from X to L_2 on the half space as the well-known Carleson-type space of functions introduced by Dahlberg. We also extend these results to L_p generalizations of the space X. Our results elaborate on the well-known duality between Carleson measures and non-tangential maximal functions.

Potential estimates and gradient boundedness for nonlinear parabolic systems

We consider a class of parabolic systems and equations in divergence form modeled by the evolutionary p -Laplacean system u_t - \operatorname{div} (|Du|^{p-2}Du)=V(x,t) , and provide L^\infty -bounds for the spatial gradient of solutions Du via nonlinear potentials of the right hand side datum V . Such estimates are related to those obtained by Kilpeläinen and Malý [22] in the elliptic case. In turn, the potential estimates found imply optimal conditions for the boundedness of Du in terms of borderline rearrangement invariant function spaces of Lorentz type. In particular, we prove that if V\in L(n+2,1) then Du \in L^\infty_{\mathrm{loc}} , where n is the space dimension, and this gives the borderline case of a result of DiBenedetto [5]; a significant point is that the condition V \in L(n+2,1) is independent of p . Moreover, we find explicit forms of local a priori estimates extending those from [5] valid for the homogeneous case V \equiv 0 .

On the reconstruction of interfaces using complex geometrical optics solutions for the acoustic case

We deal with the problem of reconstruction of the coefficient discontinuities (or supports) of scalar divergence form equations with lower order terms from the Dirichlet-to-Neumann map using complex geometrical optics (CGO) solutions. We consider both penetrable and impenetrable obstacles. The usual proofs for justifying this method assume, in addition to the smoothness of the coefficients and the interfaces, the following two conditions.The finiteness of the touching points of the phase’s level curves (or surfaces) of the used CGO solutions with the interface.The positivity of the lower bound of the relative curvature of the interface.In this paper, we show how we can remove these two conditions and justify the reconstruction method considering L∞ coefficients and Lipschitz interfaces of discontinuity.

An Lp‐Estimate for Weak Solutions of Elliptic Equations

We prove an Lp‐a priori bound, p > 2, for solutions of second order linear elliptic partial differential equations in divergence form with discontinuous coefficients in unbounded domains.

On a Condition for the Nonexistence of W-Solutions of Nonlinear High-Order Equations with L¹ -Data

In a bounded open set of R n we consider the Dirichlet problem for nonlinear2m-order equations in divergence form with L 1 -right-hand sides. It is supposed that 2 6m 0 with respect to the derivatives of orderm of unknown function. We establish that under the condition p 6 2 -m/n for some L 1 -data the corresponding Dirichlet problem does not have W-solutions. RESUMEN