December 12, 2012

Record 1 of 5.

Click Here to View Full Record

Title:

A REFINED RESULT ON SIGN CHANGING SOLUTIONS FOR A CRITICAL ELLIPTIC PROBLEM

Author Full Names:

Ge, Yuxin; Musso, Monica; Pistoia, Angela; Pollack, Daniel

Source:

COMMUNICATIONS ON PURE AND APPLIED ANALYSIS, 12 (1):125-155; 10.3934/cpaa.2013.12.125 JAN 2013

Abstract:

In this work, we consider sign changing solutions to the critical elliptic problem Delta u + vertical bar u vertical bar 4/N-2 u = 0 in Omega(epsilon) and u = 0 on partial derivative Omega(epsilon), where Omega(epsilon) := Omega - (boolean OR(m)(i=1) (a(i) + epsilon Omega(i))) for small parameter epsilon > 0 is a perforated domain, Omega and Omega(i) with 0 is an element of Omega(i) (for all(i) = 1; ... ; m) are bounded regular general domains without symmetry in R-N and a(i) are points in Omega for all i = 1, ... , m. As epsilon goes to zero, we construct by gluing method solutions with multiple blow up at each point a(i) for all i - 1, ... , m

Record 2 of 5.

Click Here to View Full Record

Title:

Strichartz estimates and the nonlinear Schrodinger equation on manifolds with boundary

Author Full Names:

Blair, Matthew D.; Smith, Hart F.; Sogge, Chris D.

Source:

MATHEMATISCHE ANNALEN, 354 (4):1397-1430; 10.1007/s00208-011-0772-y DEC 2012

Abstract:

We establish Strichartz estimates for the Schrodinger equation on Riemannian manifolds (Omega, g) with boundary, for both the compact case and the case that Omega is the exterior of a smooth, non-trapping obstacle in Euclidean space. The estimates for exterior domains are scale invariant; the range of Lebesgue exponents (p, q) for which we obtain these estimates is smaller than the range known for Euclidean space, but includes the key estimate, which we use to give a simple proof of well-posedness results for the energy critical Schrodinger equation in 3 dimensions. Our estimates on compact manifolds involve a loss of derivatives with respect to the scale invariant index. We use these to establish well-posedness for finite energy data of certain semilinear Schrodinger equations on general compact manifolds with boundary.

Record 3 of 5.

Click Here to View Full Record

Title:

Interacting superprocesses with discontinuous spatial motion

Author Full Names:

Chen, Zhen-Qing; Wang, Hao; Xiong, Jie

Source:

FORUM MATHEMATICUM, 24 (6):1183-1223; 10.1515/FORM.2011.100 NOV 2012

Abstract:

A class of interacting superprocesses arising from branching particle systems with continuous spatial motions, called superprocesses with dependent spatial motion (SDSMs), has been introduced and studied by Wang and by Dawson, Li and Wang. In this paper, we extend the model to allow discontinuous spatial motions. Under Lipschitz condition for coefficients, we show that under a proper rescaling, branching particle systems with jump-diffusion underlying motions in a random medium converge to a measure-valued process, called stable SDSM. We further characterize this stable SDSM as a unique solution of a well-posed martingale problem. To prove the uniqueness of the martingale problem, we establish the C2+gamma-regularity for the transition semigroup of a class of jump-diffusion processes, which may be of independent interest.

Record 4 of 5.

Click Here to View Full Record

Title:

Quantitative photo-acoustic tomography with partial data

Author Full Names:

Chen, Jie; Yang, Yang

Source:

INVERSE PROBLEMS, 28 (11):10.1088/0266-5611/28/11/115014 NOV 2012

Abstract:

Photo-acoustic tomography is a newly developed hybrid imaging modality that combines a high-resolutionmodality with a high-contrast modality. We analyze the reconstruction of diffusion and absorption parameters in an elliptic equation and extend an earlier result of Bal and Uhlmann (2010 Inverse Problems 26 085010) to the partial data case. We show that the reconstruction can be uniquely determined by the knowledge of four internal data based on well-chosen partial boundary conditions. Stability of this reconstruction is ensured if a convexity condition is satisfied. A similar stability result is obtained without this geometric constraint if 4n well chosen partial boundary conditions are available, where n is the spatial dimension. The set of well chosen boundary measurements is characterized by some complex geometric optics solutions vanishing on a part of the boundary.

Record 5 of 5.

Click Here to View Full Record

Title:

On subharmonicity for symmetric Markov processes

Author Full Names:

Chen, Zhen-Qing; Kuwae, Kazuhiro

Source:

JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN, 64 (4):1181-1209; 10.2969/jmsj/06441181 OCT 2012

Abstract:

We establish the equivalence of the analytic and probabilistic notions of subharmonicity in the framework of general symmetric Hunt processes on locally compact separable metric spaces, extending an earlier work of the first named author on the equivalence of the analytic and probabilistic notions of harmonicity. As a corollary, we prove a strong maximum principle for locally bounded finely continuous subharmonic functions in the space of functions locally in the domain of the Dirichlet form under some natural conditions.