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1.
We show that the free boundary ∂{u > 0}, arising from the minimizer(s) u, of the functional
approaches the (smooth) fixed boundary ∂Ω tangentially, at points where the Dirichlet data vanishes along with its gradient.   相似文献   

2.
In this paper we consider the following two-phase obstacle-problem-like equation in the unit half-ball
$\Delta u = \lambda_{+} \chi_{\{u >0 \}}-\lambda_{-} \chi_{\{u <0 \}},\quad \lambda_\pm >0 .$\Delta u = \lambda_{+} \chi_{\{u >0 \}}-\lambda_{-} \chi_{\{u <0 \}},\quad \lambda_\pm >0 .  相似文献   

3.
In this paper we introduce a two phase version of the well-known Quadrature Domain theory, which is a generalized (sub)mean value property for (sub)harmonic functions. In concrete terms, and after reformulation into its PDE version the problem boils down to finding solution to
$ - \Delta u = (\mu_+ - \lambda_+ )\chi_{\{u > 0\}} - (\mu_- - \lambda_- )\chi_{\{u < 0\}} ~~~{\rm in }~~~ {I\!\!R}^N. $ - \Delta u = (\mu_+ - \lambda_+ )\chi_{\{u > 0\}} - (\mu_- - \lambda_- )\chi_{\{u < 0\}} ~~~{\rm in }~~~ {I\!\!R}^N.  相似文献   

4.
In this paper we are concerned with singular points of solutions to the unstable free boundary problem
$\Delta u = - \chi_{\{u>0\}} \quad\hbox{in } B_1.$\Delta u = - \chi_{\{u>0\}} \quad\hbox{in } B_1.  相似文献   

5.
Let M be an n-dimensional connected compact manifold with non-empty boundary equipped with a Riemannian metric g, a spin structure σ and a chirality operator Γ. We define and study some properties of a spin conformal invariant given by:
where is the smallest eigenvalue of the Dirac operator under the chiral bag boundary condition . More precisely, we show that if n ≥ 2 then:
  相似文献   

6.
Concerning the obstacle-problem-like equation , where + > 0 and – > 0, we give a complete characterization of all global two-phase solutions with quadratic growth both at 0 and infinity.  相似文献   

7.
Let Ω be an open bounded domain in with smooth boundary . We are concerned with the critical Neumann problem
where and Q(x) is a positive continuous function on . Using Moser iteration, we give an asymptotic characterization of solutions for (*) at the origin. Under some conditions on Q,  μ, we, by means of a variational method, prove that there exists such that for every , problem (*) has a positive solution and a pair of sign-changing solutions.  相似文献   

8.
We establish the optimal regularity (of class W ∞ 2 ) of a solution to the two-phase obstacle problem
with a nonhomogeneous Dirichlet condition in a bounded domain Ω ⊂ ℝn with smooth boundary ∂Ω. Bibliography: 10 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 34, 2006, pp. 3–11.  相似文献   

9.
We investigate the boundary growth of positive superharmonic functions u on a bounded domain Ω in , n ≥ 3, satisfying the nonlinear elliptic inequality
where c >  0, α ≥ 0 and p >  0 are constants, and is the distance from x to the boundary of Ω. The result is applied to show a Harnack inequality for such superharmonic functions. Also, we study the existence of positive solutions, with singularity on the boundary, of the nonlinear elliptic equation
where V and f are Borel measurable functions conditioned by the generalized Kato class.  相似文献   

10.
We study the existence of different types of positive solutions to problem
where , , and is the critical Sobolev exponent. A careful analysis of the behavior of Palais-Smale sequences is performed to recover compactness for some ranges of energy levels and to prove the existence of ground state solutions and mountain pass critical points of the associated functional on the Nehari manifold. A variational perturbative method is also used to study the existence of a non trivial manifold of positive solutions which bifurcates from the manifold of solutions to the uncoupled system corresponding to the unperturbed problem obtained for ν = 0. B. Abdellaoui and I. Peral supported by projects MTM2007-65018, MEC and CCG06-UAM/ESP-0340, Spain. V. Felli supported by Italy MIUR, national project Variational Methods and Nonlinear Differential Equations.  相似文献   

11.
We consider the following Liouville equation in
For each fixed and a j  > 0 for 1 ≤ j ≤ k, we construct a solution to the above equation with the following asymptotic behavior:
  相似文献   

12.
Let us consider the linear boundary value problem
((0.1))
where
and
is defined by
Classical Lyapunov inequality states that
for any function
where
The constant 4/L is optimal. Let us note that Lyapunov inequality is given in terms of
the usual norm in the space L1(0, L). In this paper we review some recent results on Lp Lyapunovtype inequalities,
, for ordinary and partial differential equations on a bounded and regular domain in
In the last case, it is showed that the relation between the quantities p and N/2 plays a crucial role, pointing out a deep difference with respect to the ordinary case. In the proof, the best constants are obtained by using a related variational problem and Lagrange multiplier theorem. Finally, the linear results are combined with Schauder fixed point theorem in the study of resonant nonlinear problems. The authors have been supported by the Ministry of Science and Technology of Spain MTM2005- 01331 and by Junta de Andalucia (FQM116).  相似文献   

13.
This paper deals with nonnegative solutions of for
with and prescribed continuous Dirichlet data B = B(x) on ∂Ω. It is proved that for n ≤ 6 there is a critical parameter with the following property: If q >  q c then there exist at least two continuous weak solutions emanating from some explicitly known stationary solution w: one that coincides with w and another one that satisfies u ≥ w but . For n ≤ 6 and q ≤ q c (or n ≥ 7), however, such a second solution above w is impossible. Moreover, it is shown that for n ≤ 6, q >  q c and any sufficiently small nonnegative boundary data B there exist initial values admitting at least two continuous weak solutions of (Q). The final result asserts that for any n and q nonuniqueness for (Q) holds at least for some boundary and initial data.  相似文献   

14.
We study the existence and uniqueness of solutions of the convective–diffusive elliptic equation
posed in a bounded domain , with pure Neumann boundary conditions
Under the assumption that with p = N if N ≥ 3 (resp. p > 2 if N  =  2), we prove that the problem has a solution if ∫Ω f dx  = 0, and also that the kernel is generated by a function , unique up to a multiplicative constant, which satisfies a.e. on Ω. We also prove that the equation
has a unique solution for all ν > 0 and the map is an isomorphism of the respective spaces. The study is made in parallel with the dual problem, with equation
The dependence on the data is also examined, and we give applications to solutions of nonlinear elliptic PDE with measure data and to parabolic problems.  相似文献   

15.
Given a nontrivial Borel measure on ℝ, let p n be the corresponding orthonormal polynomial of degree n whose zeros are λ j (n), j=1,…,n. Then for each j=1,…,n,
with
defines a discrete probability distribution. The Shannon entropy of the sequence {p n } is consequently defined as
In the case of Chebyshev polynomials of the first and second kinds, an explicit and closed formula for is obtained, revealing interesting connections with number theory. In addition, several results of numerical computations exemplifying the behavior of for other families are presented.   相似文献   

16.
In this paper, by introducing a new operator, improving and generating a p-Laplace operator for some p > 1, we consider the existence of triple positive solutions for some nonlinear m-point boundary value problems on the half-line
where is the increasing homeomorphism and positive homomorphism and . We show the existence of at least three positive solutions with suitable growth conditions imposed on the nonlinear term by using the five functionals fixed-point theorem. Project supported by Foundation of Major Project of Science and Technology of Chinese Education Ministry, SRFDP of Higher Education, NSF of Education Committee of Jiangsu Province and Project of Graduate Education Innovation of Jiangsu Province.  相似文献   

17.
  相似文献   

18.
Let be the generalized integers n j associated with a set of generalized primes p i in Beurling’s sense. On the basis of the general mean-value theorems, established in our previous work, for multiplicative function f(n j ) defined on , we prove extensions, in functional form and in mean-value form, of the Elliott–Daboussi theorem to high order mean-values. For the main result, let α,ρ, and τ be positive real constants such that α > 1,ρ≥1 and . Then a multiplicative function f satisfies the following conditions, with some constant , (1) All four series
converge and (2)
if and only if the order τρ mean-value
exists with and the limit
exists with . The proof is deduced from an intrinsic connection between m f and . An erratum to this article can be found at  相似文献   

19.
We study C 2,1 nonnegative solutions u(x,t) of the nonlinear parabolic inequalities
in a punctured neighborhood of the origin in , when and . We show that a necessary and sufficient condition on λ for such solutions u to satisfy an a priori bound near the origin is , and in this case, the a priori bound on u is
This a priori bound for u can be improved by imposing an upper bound on the initial condition of u.  相似文献   

20.
We present several sharp inequalities for the volume of the unit ball in ,
. One of our theorems states that the double-inequality
holds for all n ≥ 2 with the best possible constants
This refines and complements a result of Klain and Rota.   相似文献   

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