A Comparison Principle for a Sobolev Gradient Semi-Flow Export

@article{citeulike:5938309, abstract = {We consider gradient descent equations for energy functionals of the type S(u) = 1/2 \< u(x), A(x)u(x) \>\_{L^2} + \int\_{\Omega} V(x,u) dx, where A is a uniformly elliptic operator of order 2, with smooth coefficients. The gradient descent equation for such a functional depends on the metric under consideration. We consider the steepest descent equation for S where the gradient is an element of the Sobolev space H^{\beta}, \beta \in (0,1), with a metric that depends on A and a positive number \gamma \> \sup |V\_{22}|. We prove a weak comparison principle for such a gradient flow. We extend our methods to the case where A is a fractional power of an elliptic operator. We provide an application to the Aubry-Mather theory for partial differential equations and pseudo-differential equations by finding plane-like minimizers of the energy functional}, archivePrefix = {arXiv}, author = {Blass, Timothy and Rafael de la Llave and Valdinoci, Enrico}, citeulike-article-id = {5938309}, citeulike-linkout-0 = {http://arxiv.org/abs/0910.2214}, citeulike-linkout-1 = {http://arxiv.org/pdf/0910.2214}, day = {12}, eprint = {0910.2214}, keywords = {sobolev, transport}, month = {Oct}, posted-at = {2009-11-06 15:48:24}, priority = {2}, title = {A Comparison Principle for a Sobolev Gradient Semi-Flow}, url = {http://arxiv.org/abs/0910.2214}, year = {2009} }

TY - JOUR ID - citeulike:5938309 L3 - citeulike-article-id:5938309 N2 - We consider gradient descent equations for energy functionals of the type S(u) = 1/2 < u(x), A(x)u(x) >_{L^2} + \int_{\Omega} V(x,u) dx, where A is a uniformly elliptic operator of order 2, with smooth coefficients. The gradient descent equation for such a functional depends on the metric under consideration. We consider the steepest descent equation for S where the gradient is an element of the Sobolev space H^{\beta}, \beta \in (0,1), with a metric that depends on A and a positive number \gamma > \sup |V_{22}|. We prove a weak comparison principle for such a gradient flow. We extend our methods to the case where A is a fractional power of an elliptic operator. We provide an application to the Aubry-Mather theory for partial differential equations and pseudo-differential equations by finding plane-like minimizers of the energy functional TI - A Comparison Principle for a Sobolev Gradient Semi-Flow KW - sobolev KW - transport AU - Blass, Timothy AU - Rafael de la Llave AU - Valdinoci, Enrico PY - 2009/Oct/12/ UR - http://arxiv.org/abs/0910.2214 ER -