Elliptic and parabolic obtacle problems with irregular obstacles

Principal investigator: Frank Duzaar, Verena Bögelein
Staff member: André Erhardt

The projekt was funded 2010-2014 by the DFG (Deutsche Forschungsgemeinschaft) at the University of Erlangen-Nürnberg.

Description of the project: On the one hand, the aim of the project is the development of a Calderón & Zygmund theory for solving elliptic and parabolic obstacle problems for partial differential operators in divergence form of p-Laplace type, and, on the other hand, the derivation of pointwise potential estimates of the solutions in terms of the obstacle. The objective is to achieve a proof of a classic Calderón & Zygmund estimate for the spatial gradients of the solution in terms of the integrability of the obstacle. More specifically, we want to show that the gradient is just as integrable as the obstacle function. In doing so, very low regularity requirements will be placed on the definitive vector field of the differential operators. Moreover, obstacle functions which do not necessarily diminish in time are also to be considered. Furthermore, potential estimates for the solution and their gradients in terms of a nonlinear Wolff potential of the obstacle function are to be deduced. The stationary as well as the non-stationary cases are to be considered here, whereupon the non-stationary case will likely be restricted to differential operators with linear growth.

Publications

• A. Erhardt. Existence of solutions to parabolic problems with nonstandard growth and irregular obstacles. Adv. Differential Equations 21:463-504, 2016. 
• A. Erhardt. Higher integrability for solutions to parabolic problems with irregular obtacles and nonstandard growth. J. Math. Anal. Appl. 435:1772-1803, 2016. 
• V. Bögelein, T. Lukkari and C. Scheven. The obstacle problem for the porous medium equation. Math. Ann. 363(1):455-499, 2015.
• A. Erhardt, Hölder estimates for parabolic obstacle problems. Ann. Mat. Pura Appl. (4) 194:645-671, 2015.
• C. Scheven, Existence of localizable solutions to nonlinear parabolic problems with irregular obstacles. Manuscripta Math. 146(1-2):7-63, 2015.
• P. Baroni and V. Bögelein. Calderón-Zygmund estimates for parabolic p(x,t)-Laplacian systems. Rev. Mat. Iberoam. 30(4):1355-1386, 2014.
• A. Erhardt, Calderón-Zygmund theory for parabolic obstacle problems with non- standard growth. Adv. Nonlinear Anal., 3:15-44, 2014.
• A. Erhardt. Existence and Gradient Estimates in Parabolic Obstacle Problems with Nonstandard Growth. Dissertation, 2013.
• V. Bögelein and C. Scheven, Higher integrability in parabolic obstacle problems. Forum Math, 24(5):931–972, 2012.
• C. Scheven, Elliptic obstacle problems with measure data: Potentials and low order regularity. Publ. Mat. 56(2):327–374, 2012.
• C. Scheven, Gradient potential estimates in non-linear elliptic obstacle problems with measure data. J. Func. Anal. 262(6):2777–2832, 2012.
• C. Scheven, Potential estimates in parabolic obstacle problems. Ann. Acad. Sci. Fenn. Math. 37:415–443, 2012.
• V. Bögelein and F. Duzaar, Higher integrability for parabolic systems with non-standard growth and degenerate diffusions. Publ. Mat., 55(1):201–250, 2011.
• C. Scheven, Existence and Gradient Estimates in Nonlinear Problems with Irregular Obstacles. Habilitation, 2011.