共查询到20条相似文献,搜索用时 31 毫秒
1.
Jason J. Sharples 《Journal of Differential Equations》2004,202(1):111-142
We consider linear parabolic equations of second order in a Sobolev space setting. We obtain existence and uniqueness results for such equations on a closed two-dimensional manifold, with minimal assumptions about the regularity of the coefficients of the elliptic operator. In particular, we derive a priori estimates relating the Sobolev regularity of the coefficients of the elliptic operator to that of the solution. The results obtained are used in conjunction with an iteration argument to yield existence results for quasilinear parabolic equations. 相似文献
2.
YeMinCHEN 《数学学报(英文版)》2004,20(6):1103-1118
The aim of this paper is to study the regularity of solutions to the Dirichlet problems for general second-order elliptic equations in Lebesgue and Morrey spaces. We consider both nondivergence and divergence forms and the coefficients of principle terms are assumed to be in VMO. 相似文献
3.
In this paper we obtain the global regularity estimates of the weak solutions in Sobolev spaces and Orlicz spaces for higher order elliptic and parabolic equations of divergence form with small BMO coefficients in the whole space. We only focus on the parabolic case while the corresponding result in the elliptic case can be obtained as a corollary. 相似文献
4.
We establish global regularity for weak solutions to quasilinear divergence form elliptic and parabolic equations over Lipschitz domains with controlled growth conditions on low order terms. The leading coefficients belong to the class of BMO functions with small mean oscillations with respect to x. 相似文献
5.
We present some general methods for the estimation of the local Hausdorff measure of nodal sets of solutions to elliptic and parabolic equations. Our main results (Theorems 3.1 and 4.1) improve previous results of Lin Fanghua in [1]. 相似文献
6.
We examine the bifurcations to positive and sign-changing solutions of degenerate elliptic equations. In the problems we study, which do not represent Fredholm operators, we show that there is a critical parameter value at which an infinity of bifurcations occur from the trivial solution. Moreover, a bifurcation occurs at each point in some unbounded interval in parameter space. We apply our results to non-monotone eigenvalue problems, degenerate semi-linear elliptic equations, boundary value differential-algebraic equations and fully non-linear elliptic equations.
7.
Zeev Schuss 《Journal of Mathematical Analysis and Applications》1973,44(1):136-159
We consider a parabolic system in a half space. A theorem, similar to one proved by Meyers and Pazy for elliptic equations outside the unit ball is proved, namely, if the coefficients, the right side, and the initial conditions of the parabolic system have asymptotic expansions at infinity with respect to the space variable, then so does the solution of the corresponding Cauchy problem. Some generalizations and examples are given. 相似文献
8.
We study a forward-backward system
of stochastic differential equations in an
infinite-dimensional framework and its relationships
with a semilinear parabolic differential equation on a Hilbert space,
in the spirit of the approach of Pardoux-Peng.
We prove that the stochastic system
allows us to construct a unique
solution of the parabolic equation in
a suitable class of locally Lipschitz real
functions. The parabolic equation is understood in
a mild sense which requires the notion
of a generalized directional gradient, that
we introduce by a probabilistic approach
and prove to exist for locally Lipschitz
functions.
The use of the generalized directional gradient
allows us to cover various applications to option
pricing problems and to optimal stochastic control problems
(including control of delay equations and
reaction--diffusion equations),
where the lack of differentiability of the coefficients
precludes differentiability of solutions to the associated
parabolic equations of Black--Scholes or Hamilton-Jacobi-Bellman
type. 相似文献
9.
N.V. Krylov 《Journal of Functional Analysis》2009,257(6):1695-3327
The solvability in spaces is proved for second-order elliptic equations with coefficients which are measurable in one direction and VMO in the orthogonal directions in each small ball with the direction depending on the ball. This generalizes to a very large extent the case of equations with continuous or VMO coefficients. 相似文献
10.
11.
Fabio Punzo 《Journal of Differential Equations》2011,251(7):1972-1989
We investigate existence and uniqueness of solutions to semilinear parabolic and elliptic equations in bounded domains of the n-dimensional hyperbolic space (n?3). Lp→Lq estimates for the semigroup generated by the Laplace-Beltrami operator are obtained and then used to prove existence and uniqueness results for parabolic problems. Moreover, under proper assumptions on the nonlinear function, we establish uniqueness of positive classical solutions and nonuniqueness of singular solutions of the elliptic problem; furthermore, for the corresponding semilinear parabolic problem, nonuniqueness of weak solutions is stated. 相似文献
12.
Global solutions for quasilinear parabolic problems 总被引:4,自引:0,他引:4
Results on the global existence of classical solutions for quasilinear parabolic equations in bounded domains with homogeneous
Dirichlet or Neumann boundary conditions are presented. Besides quasilinear parabolic equations, the method is also applicable
to some weakly-coupled reaction-diffusion systems and to elliptic equations with nonlinear dynamic boundary conditions.
Received December 21, 2000; accepted August 30, 2001. 相似文献
13.
Roberto Stasi 《Potential Analysis》2007,26(3):213-224
In this paper we prove the validity of the Maximum Principle for some class of elliptic and parabolic equations of diffusion
type in infinite dimension. The main tools are Asplund’s theorem and Preiss’ theorem on differentiability of Lipschitz functions
in Banach space.
相似文献
14.
N. V. Krylov 《偏微分方程通讯》2013,38(3):453-475
An Lp-theory of divergence and non-divergence form elliptic and parabolic equations is presented. The main coefficients are supposed to belong to the class VMOx, which, in particular, contains all functions independent of x. Weak uniqueness of the martingale problem associated with such equations is obtained. 相似文献
15.
Qi Zhang 《Transactions of the American Mathematical Society》1996,348(7):2811-2844
We obtain the existence of the weak Green's functions of parabolic equations with lower order coefficients in the so called parabolic Kato class which is being proposed as a natural generalization of the Kato class in the study of elliptic equations. As a consequence we are able to prove the existence of solutions of some initial boundary value problems. Moreover, based on a lower and an upper bound of the Green's function, we prove a Harnack inequality for the non-negative weak solutions.
16.
We consider both divergence and non-divergence parabolic equations on a half space in weighted Sobolev spaces. All the leading coefficients are assumed to be only measurable in the time and one spatial variable except one coefficient, which is assumed to be only measurable either in the time or the spatial variable. As functions of the other variables the coefficients have small bounded mean oscillation (BMO) semi-norms. The lower-order coefficients are allowed to blow up near the boundary with a certain optimal growth condition. As a corollary, we also obtain the corresponding results for elliptic equations. 相似文献
17.
Jessica Lin 《偏微分方程通讯》2015,40(9):1688-1704
This note establishes an interior quantitative lower bound for nonnegative supersolutions of fully nonlinear uniformly parabolic equations. The result may be interpreted as a quantitative version of a growth lemma established by Krylov and Safonov for nonnegative supersolutions of linear uniformly parabolic equations in nondivergence form. Our approach is different, and follows from an application of a reverse Holder inequality. The result is the parabolic analogue of an elliptic regularity estimate established by Caffarelli, Souganidis, and Wang in the stochastic homogenization of fully nonlinear uniformly elliptic equations. 相似文献
18.
Fabio Punzo 《Mathematische Nachrichten》2013,286(10):1043-1054
We investigate uniqueness for degenerate parabolic and elliptic equations in the class of solutions belonging to weighted Lebesgue spaces and not satisfying any boundary condition. The uniqueness result that we provide relies on the existence of suitable positive supersolutions of the adjoint equations. Under proper assumptions on the behavior at the boundary of the coefficients of the operator, such supersolutions are constructed, mainly using the distance function from the boundary. 相似文献
19.
We study some semilinear elliptic equations with singular coefficients which relate to some Hardy–Sobolev inequalities. We obtain some existence results for these equations and give a theorem for prescribing the Palais–Smale sequence for these equations. Moreover, we find some interesting connections between these equations and some semilinear elliptic equations in hyperbolic space. Using these connections, we obtain many new results for these equations. 相似文献
20.
László Simon 《Periodica Mathematica Hungarica》2008,56(1):143-156
We consider a system consisting of a quasilinear parabolic equation and a first order ordinary differential equation where
both equations contain functional dependence on the unknown functions. Then we consider a system which consists of a quasilinear
parabolic partial differential equation, a first order ordinary differential equation and an elliptic partial differential
equation. These systems were motivated by models describing diffusion and transport in porous media with variable porosity.
Supported by the Hungarian NFSR under grant OTKA T 049819. 相似文献