arXiv subjects
Publications and source records attributed to Mohammad Safdari.
We study the regularity of the distance function to the boundary of a domain in $\mathbb{R}^n$, with respect to the Minkowski functional of a convex polytope. We obtain the regularity of the distance function in certain cases. We also explicitly compute the distance function in a collection of examples and observe the new interesting phenomena that arise for such distance functions.
We study temperature distribution in a heat conducting problem, for a system of p-Laplace equation, giving rise to a free boundary.
We prove the existence and $C^{1,\alpha}$ regularity of solutions to nonlocal fully nonlinear elliptic equations with gradient constraints. We do not assume any regularity about the constraints; so the constraints need not be $C^1$ or strictly convex. We also obtain $C^{0,1}$ boundary regularity for these problems. Our approach is to show that these nonlocal equations with gradient constraints are related to some nonlocal double obstacle problems. Then we prove the regularity of the double obstacle problems. In this process, we also employ the monotonicity property for the second derivative of obstacles, which we have obtained in a previous work.
We prove the existence and $C^{1,α}$ regularity of solutions to nonlocal fully nonlinear elliptic double obstacle problems. We also obtain boundary regularity for these problems. The obstacles are assumed to be Lipschitz semi-concave/semi-convex functions, and we do not require them to be $C^1$. Our approach is to adapt a penalization method to be applicable to the setting of nonlocal equations and their viscosity solutions.
We prove the optimal $W^{2, \infty }$ regularity for fully nonlinear elliptic equations with convex gradient constraints. We do not assume any regularity about the constraints; so the constraints need not be $C^1$ or strictly convex. We also show that the optimal regularity holds up to the boundary. Our approach is to show that these elliptic equations with gradient constraints are related to some fully nonlinear double obstacle problems. Then we prove the optimal $W^{2, \infty }$ regularity for the double obstacle problems. In this process, we also employ the monotonicity property for the second derivative of obstacles, which we have obtained in a previous work.
We prove the optimal $W^{2,\infty}$ regularity for variational problems with convex gradient constraints. We do not assume any regularity of the constraints; so the constraints can be nonsmooth, and they need not be strictly convex. When the domain is smooth enough, we show that the optimal regularity holds up to the boundary. In this process, we also characterize the set of singular points of the viscosity solutions to some Hamilton-Jacobi equations. Furthermore, we obtain an explicit formula for the second derivative of these viscosity solutions; and we show that the second derivatives satisfy a monotonicity property.
For an evolution of metrics $(M,g_{t})$ there is a t-smooth family of embeddings $e_{t}:M\to\mathbb{R}^{N}$ inducing $g_{t}$, but in general there is no family of embeddings extending a given initial embedding $e_{0}$. We give an example of this phenomenon when $g_{t}$ is the evolution of $g_{0}$ under the Ricci flow. We show that there are embeddings $e_{0}$ inducing $g_{0}$ which do not admit of t-smooth extensions to $e_{t}$ inducing $g_{t}$ for any t>0. We also find hypersurfaces of dim>2 that will not remain a hypersurface under Ricci flow for any positive time.
We study the regularity of the distance function to the boundary of a domain in $\mathbb{R}^2$, with respect to some asymmetric norms. We allow the boundary of the domain to have corners. We obtain an explicit formula for the second derivative of these distance functions. Furthermore, we study a generalized notion of the ridge of a domain, which is the set of singularities of a distance function to the boundary of the domain. We completely characterize the ridge by using a generalized notion of curvature.
In this paper we prove that the free boundary of some variational inequalities with gradient constraints is as regular as the tangent bundle of the boundary of the domain. To this end, we study a generalized notion of ridge of a domain in the plane, which is the set of singularities of the distance function in the p-norm to the boundary of the domain.
In this paper we derive an estimate on the number of local maxima of the free boundary of some variational inequalities with pointwise gradient constraints. This also gives an estimate on the number of connected components of the free boundary.
We prove the optimal regularity for some class of vector-valued variational inequalities with gradient constraints. We also give a new proof for the optimal regularity of some scalar variational inequalities with gradient constraints. In addition, we prove that some class of variational inequalities with gradient constraints are equivalent to an obstacle problem, both in the scalar and vector-valued case.