共查询到20条相似文献,搜索用时 31 毫秒
1.
Sun-Sig Byun 《Journal of Evolution Equations》2007,7(3):415-428
This work treats Lp regularity theory for weak solutions of parabolic equations in divergence form with discontinuous coefficients on nonsmooth
domains. We essentially obtain an optimal condition on the coefficients under which the global W1,p regularity theory holds.
This work was supported by SNU foundation in 2005. 相似文献
2.
We investigate entire radial solutions of the semilinear biharmonic equation Δ2u=λexp(u) in Rn, n?5, λ>0 being a parameter. We show that singular radial solutions of the corresponding Dirichlet problem in the unit ball cannot be extended as solutions of the equation to the whole of Rn. In particular, they cannot be expanded as power series in the natural variable s=log|x|. Next, we prove the existence of infinitely many entire regular radial solutions. They all diverge to −∞ as |x|→∞ and we specify their asymptotic behaviour. As in the case with power-type nonlinearities [F. Gazzola, H.-Ch. Grunau, Radial entire solutions for supercritical biharmonic equations, Math. Ann. 334 (2006) 905-936], the entire singular solution x?−4log|x| plays the role of a separatrix in the bifurcation picture. Finally, a technique for the computer assisted study of a broad class of equations is developed. It is applied to obtain a computer assisted proof of the underlying dynamical behaviour for the bifurcation diagram of a corresponding autonomous system of ODEs, in the case n=5. 相似文献
3.
Flaviano Battelli Michal Fečkan Matteo Franca 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2008,59(2):212-243
We study the problem of existence of periodic solutions to a partial differential equation modelling the behavior of an undamped
beam subject to an external periodic force. We assume that the ordinary differential equation associated to the first two
modes of vibration of the beam has a symmetric homoclinic solution. By using methods borrowed by dynamical systems theory
we prove that, if the period is non resonant with the (infinitely many) internal periods of the PDE, the equation has a weak
periodic solution of the same period as the external force. In particular we obtain continua of periodic solutions for the
undamped beam in absence of external forces. This result may be considered as an infinite dimensional analogue of a result
obtained in [16] concerning accumulation of periodic solutions to homoclinic orbits in finite dimensional reversible systems.
Matteo Franca: Partially supported by G.N.A.M.P.A. – INdAM (Italy). 相似文献
4.
In this article, we prove that there are three positive solutions of a semilinear elliptic equation in a bounded symmetric domain t for large t>0 in which one is axially symmetric and the other two are nonaxially symmetric. Main tools are the Palais-Smale theory. 相似文献
5.
In this paper we study the existence of bounded weak solutions for some nonlinear Dirichlet problems in unbounded domains. The principal part of the operator behaves like the p-laplacian operator, and the lower order terms, which depend on the solution u and its gradient u, have a power growth of order p–1 with respect to these variables, while they are bounded in the x variable. The source term belongs to a Lebesgue space with a prescribed asymptotic behaviour at infinity. 相似文献
6.
An extended theory for elastic and plastic beam problems is studied. By introducing new dependent and independent variables, the standard Timoshenko beam model is extended to take account of shear variation in the lateral direction. The dynamic governing equations are established via Hamilton's principle, and existence and uniqueness results for the solution of the static problem are proved. Using the theory of convex analysis, the duality theory for the extended beam model is developed. Moreover, the extended theory for rigid-perfectly plastic beams is also established. Based on the extended model, a finite-element method is proposed and numerical results are obtained indicating the usefulness of the extended theory in applications.The work of the first author was supported in part by National Science Foundation under Grant DMS9400565. 相似文献
7.
We study the existence of weak solutions for stationary viscous fluids with general slip boundary conditions in this paper. Applying monotone operator theory, we first establish the existence result of weak solutions for an approximation problem. Then using the compactness methods and the point-wise convergence property of velocity gradients, we get the desired results. 相似文献
8.
We study a nonlinear eigenvalue problem with a nonsmooth potential. The subgradients of the potential are only positive near
the origin (from above) and near +∞. Also the subdifferential is not necessarily monotone (i.e. the potential is not convex).
Using variational techniques and the method of upper and lower solutions, we establish the existence of at least two strictly
positive smooth solutions for all the parameters in an interval. Our approach uses the nonsmooth critical point theory for
locally Lipschitz functions. A byproduct of our analysis is a generalization of a result of Brezis-Nirenberg (CRAS, 317 (1993))
on H10 versus C10 minimizers of a C1-functional. 相似文献
9.
We prove monotonicity formulae related to degenerate k-Hessian equations which yield Morrey type estimates for certain integrands involving the second derivatives of the solution.
In the special case k=2 we deduce that weak solutions in , , have locally H?lder continuous gradients. In the nondegenerate case we also show that weak solutions in , , have locally bounded second derivatives.
Received February 25, 1999 / Accepted June 11, 1999 / Published online April 6, 2000 相似文献
10.
Ezequiel R. Barbosa 《Journal of Differential Equations》2011,250(8):3398-3417
We consider potential elliptic systems involving p-Laplace operators, critical nonlinearities and lower-order perturbations. Suitable necessary and sufficient conditions for existence of nontrivial solutions are presented. In particular, a number of results on Brezis-Nirenberg type problems are extended in a unified framework. 相似文献
11.
Juan Manfredi Arshak Petrosyan Henrik Shahgholian 《Calculus of Variations and Partial Differential Equations》2002,14(3):359-384
We consider a free boundary problem for the p-Laplacian
describing nonlinear potential flow past a convex profile K with prescribed pressure on the free stream line. The main purpose of this paper is to study the limit as of the classical solutions of the problem above, existing under certain convexity assumptions on a(x). We show, as one can expect, that the limit solves the corresponding potential flow problem for the -Laplacian
in a certain weak sense, strong enough however, to guarantee uniqueness. We show also that in the special case the limit is given by the distance function.
Received: 10 October 2000 / Accepted: 23 February 2001 / Published online: 19 October 2001 相似文献
12.
Francesco Chiacchio Tonia Ricciardi 《NoDEA : Nonlinear Differential Equations and Applications》2007,13(5-6):563-584
We prove the existence of at least two doubly periodic vortex solutions for a self-dual CP(1) Maxwell-Chern-Simons model. To this end we analyze a system of two elliptic equations with exponential nonlinearities.
Such a system is shown to be equivalent to a fourth-order elliptic equation admitting a variational structure.
Tonia Ricciardi: Partially supported by the MIUR National Project Variational Methods and Nonlinear Differential Equations 相似文献
13.
Jorge García-Melián Julio D. Rossi José C. Sabina de Lis 《Calculus of Variations and Partial Differential Equations》2008,31(2):187-204
In this work we consider the behaviour for large values of p of the unique positive weak solution u
p
to Δ
p
u = u
q
in Ω, u = +∞ on , where q > p − 1. We take q = q(p) and analyze the limit of u
p
as p → ∞. We find that when q(p)/p → Q the behaviour strongly depends on Q. If 1 < Q < ∞ then solutions converge uniformly in compacts to a viscosity solution of with u = +∞ on . If Q = 1 then solutions go to ∞ in the whole Ω and when Q = ∞ solutions converge to 1 uniformly in compact subsets of Ω, hence the boundary blow-up is lost in the limit. 相似文献
14.
We consider a circulation system arising in turbulence modelling in fluid dynamics with unbounded eddy viscosities. Various notions of weak solution are considered and compared. We establish existence and regularity results. In particular we study the boundedness of weak solutions. We also establish an existence result for a classical solution. 相似文献
15.
16.
Roberta Filippucci 《Journal of Differential Equations》2011,250(1):572-595
In this paper we deal with noncoercive elliptic systems of divergence type, that include both the p-Laplacian and the mean curvature operator and whose right-hand sides depend also on a gradient factor. We prove that any nonnegative entire (weak) solution is necessarily constant. The main argument of our proofs is based on previous estimates, given in Filippucci (2009) [12] for elliptic inequalities. Actually, the main technique for proving the central estimate has been developed by Mitidieri and Pohozaev (2001) [23] and relies on the method of test functions. No use of comparison and maximum principles or assumptions on symmetry or behavior at infinity of the solutions are required. 相似文献
17.
We study solutions of first order partial differential relations Du∈K, where u:Ω⊂ℝ
n
→ℝ
m
is a Lipschitz map and K is a bounded set in m×n matrices, and extend Gromov’s theory of convex integration in two ways. First, we allow for additional constraints on the
minors of Du and second we replace Gromov’s P-convex hull by the (functional) rank-one convex hull. The latter can be much larger than the former and this has important
consequences for the existence of ‘wild’ solutions to elliptic systems. Our work was originally motivated by questions in
the analysis of crystal microstructure and we establish the existence of a wide class of solutions to the two-well problem
in the theory of martensite.
Received April 23, 1999 / final version received September 11, 1999 相似文献
18.
András Domokos 《Journal of Differential Equations》2004,204(2):439-470
We propose a direct method to control the first-order fractional difference quotients of solutions to quasilinear subelliptic equations in the Heisenberg group. In this way we implement iteration methods on fractional difference quotients to obtain weak differentiability in the T-direction and then second-order weak differentiability in the horizontal directions. 相似文献
19.
Yehuda Pinchover 《Mathematische Annalen》1999,314(3):555-590
In this paper we discuss some new results concerning perturbation theory for second order elliptic partial differential equations
related to positivity properties of such equations. We continue the study of some different notions of “small” perturbations
and discuss their relations to comparisons of Green's functions, refined maximum and anti-maximum principles, ground state,
and the decay of eigenfunctions.
In particular, we show that if V is a positive function which is a semismall perturbation of a subcritical Schr?dinger operator H defined on a domain , and are the (Dirichlet) eigenfunctions of the equation , then for any , the function is bounded and has a continuous extension up to the Martin boundary of the pair , where is the ground state of H with a principal eigenvalue .
Received: 29 November 1998 相似文献
20.
In this paper we give sufficient conditions for the nonexistence of positive entire weak solutions of coercive and anticoercive elliptic inequalities, both of the p-Laplacian and of the mean curvature type, depending also on u and x inside the divergence term, while a gradient factor is included on the right-hand side. In particular, to prove our theorems we use a technique developed by Mitidieri and Pohozaev in [E. Mitidieri, S.I. Pohozaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math., 234 (2001) 1–362], which relies on the method of test functions without using comparison and maximum principles. Their approach is essentially based first on a priori estimates and on the derivation of an asymptotics for the a priori estimate. Finally nonexistence of a solution is proved by contradiction. 相似文献