Stochastic variational inequalities and regularity for degenerate stochastic partial differential equations

The regularity and characterization of solutions to degenerate, quasilinear SPDE is studied. Our results are two-fold: First, we prove regularity results for solutions to certain degenerate, quasilinear SPDE driven by Lipschitz continuous noise. In particular, this provides a characterization of solutions to such SPDE in terms of (generalized) strong solutions. Second, for the one-dimensional stochastic mean curvature flow with normal noise we adapt the notion of stochastic variational inequalities to provide a characterization of solutions previously obtained in a limiting sense only. This solves a problem left open in [Es-Sarhir, von Renesse; SIAM, 2012] and sharpens regularity properties obtained in [Es-Sarhir, von Renesse, Stannat; NoDEA, 2012].

[1]  Benjamin Gess,et al.  Multi-valued, singular stochastic evolution inclusions , 2011, 1112.5672.

[2]  V. Caselles,et al.  A Parabolic Quasilinear Problem for Linear Growth Functionals , 2002 .

[3]  Xiaobing Feng,et al.  Finite element approximations of the stochastic mean curvature flow of planar curves of graphs , 2013, 1303.5930.

[4]  Panagiotis E. Souganidis,et al.  Uniqueness of motion by mean curvature perturbed by stochastic noise , 2004 .

[5]  Giorgio C. Buttazzo,et al.  Variational Analysis in Sobolev and BV Spaces - Applications to PDEs and Optimization, Second Edition , 2014, MPS-SIAM series on optimization.

[6]  Stanley Osher,et al.  Deblurring and Denoising of Images by Nonlocal Functionals , 2005, Multiscale Model. Simul..

[7]  G. Anzellotti,et al.  Pairings between measures and bounded functions and compensated compactness , 1983 .

[8]  Wilhelm Stannat,et al.  Estimates for the ergodic measure and polynomial stability of plane stochastic curve shortening flow , 2010, 1008.1961.

[9]  G. Anzellotti,et al.  The Euler equation for functionals with linear growth , 1985 .

[10]  Abdelhadi Es-Sarhir,et al.  Ergodicity of Stochastic Curve Shortening Flow in the Plane , 2010, SIAM J. Math. Anal..

[11]  M. Röckner,et al.  A Concise Course on Stochastic Partial Differential Equations , 2007 .

[12]  P. Souganidis,et al.  Fully nonlinear stochastic partial differential equations , 1998 .

[13]  Benjamin Gess,et al.  Strong Solutions for Stochastic Partial Differential Equations of Gradient Type , 2011, 1104.4243.

[14]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[15]  P. Souganidis,et al.  Fully nonlinear stochastic partial differential equations: non-smooth equations and applications , 1998 .

[16]  Feng-Yu Wang,et al.  Stochastic generalized porous media and fast diffusion equations , 2006, math/0602369.

[17]  Matteo Novaga,et al.  A stochastic selection principle in case of fattening for curvature flow , 2001 .

[18]  V. Barbu,et al.  Stochastic Variational Inequalities and Applications to the Total Variation Flow Perturbed by Linear Multiplicative Noise , 2012, 1209.0351.

[19]  Ieee Xplore,et al.  IEEE Transactions on Pattern Analysis and Machine Intelligence Information for Authors , 2022, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[20]  Donald Geman,et al.  Constrained Restoration and the Recovery of Discontinuities , 1992, IEEE Trans. Pattern Anal. Mach. Intell..

[21]  Ivan P. Gavrilyuk,et al.  Variational analysis in Sobolev and BV spaces , 2007, Math. Comput..

[22]  Arnaud Debussche,et al.  Stochastic partial differential equations: analysis and computations , 2013, Stochastic Partial Differential Equations: Analysis and Computations.

[23]  I. Gavrilyuk.  Book Review: Variational analysis in Sobolev and BV spaces , 2007 .

[24]  Panagiotis E. Souganidis,et al.  Fully nonlinear stochastic PDE with semilinear stochastic dependence , 2000 .

[25]  Giuseppe Da Prato,et al.  Existence of strong solutions for stochastic porous media equation under general monotonicity conditions , 2007, math/0703421.