The bridge principle is the idea that it should be possible to join two minimal submanifolds along their boundaries by a thin bridge and perturb the new configuration (i.e. the approximate solution) so that it is minimal. In 1987, N. Smale proved the bridge principle for smooth (possibly unstable) minimal submanifolds in Euclidean space having arbitrary dimension and codimension by solving a fixed point problem for the stability operator L on the normal bundle of the approximate solution. Two years later, Smale constructed the first examples of minimal hypersurfaces with multiple isolated singularities by extending their bridge principle to strictly stable (i.e. L positive definite) minimal hypercones in Euclidean space. We discuss a recent generalization of Smale's singular bridge principle to high codimension minimal cones. As an application, we obtain a minimal graph with any finite number of isolated singularities.
Almost-minimizers of a surface energy arise naturally in geometric variational problems. Their regularity is well understood for smooth elliptic energies, but little is known for crystalline energies outside of the small mass regime. In this talk, we will discuss a regularity result for 2-dimensional crystalline almost-minimizers.
We study the minimizers of \(\lambda_k^s(A) + |A|\) where \(\lambda^s_k(A)\) is the \(k\)-th Dirichlet eigenvalue of the fractional Laplacian on A. This free boundary problem arises from a natural generalization of the problem of minimizing the \(k\)-th eigenvalue of \(-\Delta\) and can also be considered as a vector-valued version of the classical Bernoulli-type free boundary problem. Even the local case is not fully understood, and the free boundary of minimizers in the non-local setting exhibits distinct global behavior. In this talk, I will present our three main results:
In this talk, I will present the Harnack inequality for a class of linear parabolic equations in nondivergence form in which the leading coefficients could be degenerate or singular, or both. As corollaries, weighted Hölder regularity estimates of solutions with respect to a quasi-metric, and a Liouville-type theorem will also be presented. This is a joint work with Sungwon Cho and Tuoc Phan.
We study the obstacle problem associated with the Kolmogorov operator \(\Delta_v - \partial_t - v\cdot\nabla_x\), which arises from the theory of optimal control in Asian-American options pricing models. This problem presents a significant departure from the elliptic and parabolic obstacle problems due to the highly degenerate hypoelliptic nature and the non-commutative Galilean group structure underlying the operator. Our first main result finds that solutions are Lipschitz in the \(x\)-variable, and therefore enjoy better regularity than the \(C^{1,1}\) regularity inherited from the quadratic separation from the obstacle. This is unusual in the theory of obstacle problems. Using this newfound Lipschitz regularity, we derive a new monotonicity formula and obtain the first free boundary regularity result for this problem.
In this talk I will show a novel homogenization result of the heat equation and Laplace equation with vertically oscillating, i.e., oscillating in the \(u\)-variable, Neumann boundary condition. The homogenized heat equation admits an emerging rate-independent singularly anisotropic pinned boundary motion law. The steady state equation shows similar features to and in fact generalizes the celebrated thin obstacle problem. I'll present a comparison principle for the homogenized heat equation and some regularity results for the steady state equation that generalize the classical results of Athanasopoulos-Caffarelli (2004) in this new direction. I'll also present some open questions.
We study a class of second-order parabolic and elliptic systems in divergence form with the conormal boundary condition, set in either a half-space or a bounded domain. The leading coefficients are of the form \(w(x)A^{ij}(t,x)\), where the weight \(w(x)\) may be singular or degenerate near the boundary, and \(A^{ij}(t,x)\) are strongly elliptic and satisfy a certain small mean oscillation type condition. We also consider the presence of singular lower-order terms. We obtain the well-posedness and regularity of solutions in weighted mixed-norm Sobolev spaces. This talk is based on ongoing joint work with Hongjie Dong.
The Calderón-Zygmund estimate states that the \(L^p\) norm of solutions to elliptic equations can be controlled in terms of the \(L^p\) norm for the source terms. For linear equations, it is well known that this estimate holds for any exponents \(p > 1\) but not \(p = 1\). In contrast, our main result shows the estimate holds with \(p = 1\) for a certain class of fully nonlinear equations. This talk is based on a joint work with Hongjie Dong (Brown University).