Zoom link. https://umn.zoom.us/j/99821709760

Title. Hessian Estimates for the Lagrangian mean curvature equation

Abstract. In this talk, we will derive a priori interior Hessian estimates for the Lagrangian mean curvature equation under certain natural restrictions on the Lagrangian phase. As an application, we will use these estimates to solve the Dirichlet problem for the Lagrangian mean curvature equation with continuous boundary data, on a uniformly convex, bounded domain in R^n.

Zoom link. https://umn.zoom.us/j/91467804294

Title. Additive and scalar-multiplicative Carleson perturbations of elliptic operators on domains with low dimensional boundaries.

Abstract. At the beginning of the 90s, Fefferman, Kenig and Pipher (FKP) obtained a rather sharp (additive) perturbation result for the Dirichlet problem of divergence form elliptic operators. Without delving into details, the point is that if the (additive) disagreement of two operators satisfies what is known as a Carleson measure condition, then quantitative absolute continuity of the elliptic measure is transferred from one operator to the other, if one of the operators already possesses this property. Their (additive) perturbation result has since then been generalized to increasingly weaker geometric and topological assumptions on boundaries of co-dimension 1, by multiple authors.

This talk will consist of two main parts. In the first part, we will see an extension of the FKP result to the degenerate elliptic operators of David, Feneuil and Mayboroda, which were developed to study geometric and analytic properties of sets with boundaries whose co-dimension is higher than 1. These operators are of the form -div A∇ , where A is a degenerate elliptic matrix crafted to weigh the distance to the low-dimensional boundary in a way that allows for the nourishment of an elliptic theory. When this boundary is a d-Alhfors-David regular set in R^n with d in [1, n-1), and n≥ 3, we prove that the membership of the elliptic measure in A_∞ is preserved under (additive) Carleson measure perturbations of the matrix of coefficients, yielding in turn that the L^p-solvability of the Dirichlet problem is also stable under these perturbations (with possibly different p). If the Carleson measure perturbations are suitably small, we establish solvability of the Dirichlet problem in the same L^p space. One of the corollaries of our results together with a previous result of David, Engelstein and Mayboroda, is that, given any d-ADR boundary Γ with d in [1, n-2), n≥ 3, there is a family of degenerate operators of the form described above whose elliptic measure is absolutely continuous with respect to the d-dimensional Hausdorff measure on Γ. Our method of proof uses the method of Carleson measure extrapolation, as developed by Lewis and Murray, and adapted to a dyadic setting by Hofmann and Martell in the past decade. This is joint work with Svitlana Mayboroda.

In the second part of the talk, we will adopt a slightly different perspective than has been customary in the literature of these perturbation results, by considering scalar-multiplicative Carleson perturbations, as communicated to us by Joseph Feneuil and inspired by the work on equations with drift terms of Hofmann and Lewis, and Kenig and Pipher, at the start of the 21st century. Essentially, if we may write A=bA_0 with b a scalar function bounded above and below by a positive number, and ∇b·dist(· ,Γ) satisfying a Carleson measure condition, then we still retain the transference of the quantitative absolute continuity of the elliptic measure for -div A∇, if -div A_0∇ already has this property. By way of examples in the setting of low dimensional boundaries, we will see that one ought to consider these two types of perturbations (namely, additive and scalar-multiplicative) to reckon a more complete picture of the absolute continuity of elliptic measure. This is joint work with Joseph Feneuil.

Zoom link. https://umn.zoom.us/j/93601442665

Title. On the adjacency of general dyadic grids in Euclidean spaces .

Abstract. In this talk, we will briefly introduce the recent development on study the adjacent grids in Euclidean spaces. This talk contains three 3 parts, the real line case, the higher dimension case and the different bases case. The first part is a joint work with Tess Anderson, Liwei Jiang, Connor Olson and Zeyu Wei, which is due to a summer REU project at UW-Madison in 2018; while the second and third parts are taken from joint works with Tess Anderson.

Zoom link. https://umn.zoom.us/j/96572145611

Title. Elliptic equations with singular drifts term on Lipschitz domains.

Abstract. In this talk, we consider linear elliptic equation of second-order with the first term given by a singular vector field $\mathbf{b}$ on bounded Lipschitz domains $\Omega$ in $\mathbb{R}^n$, $(n\geq 3)$. Under the assumption $\mathbf{b}\in L^n(\Omega)^n$, we establish unique solvability in $L_{\alpha}^p(\Omega)$ for Dirichlet and Neumann problems. Here $L_{\alpha}^p(\Omega)$ denotes the standard Sobolev spaces(or Bessel potential space) with the pair $(\alpha,p)$ satisfying certain condition. These results extend the classical works of Jerison-Kenig (1995) and Fabes-Mendez-Mitrea (1999) for the Poisson equation. In addition, we study the Dirichlet problem for such linear elliptic equation when the boundary data is in $L^2(\partial\Omega)$. Necessary review on this topics is also presented in this talk. This is a joint work with Hyunseok Kim(Sogang University).

Zoom link. https://umn.zoom.us/j/91393860723

Title. A sharp global Strichartz estimate for the Schrodinger equation on the cylinder

Abstract. The classical Strichartz estimates show that a solution to the linear Schrodinger equation on Euclidean space is in certain Lebesgue spaces globally in time provided the initial data is in L^2. On compact manifolds one can no longer have global control, and some loss of derivatives is necessary (meaning the initial data needs to be in a Sobolev space rather than L^2). In 'intermediate' cases it is a challenging question to understand when one can have good space-time estimates with no loss of derivatives.

In this talk we discuss a global-in-time Strichartz-type estimate for the linear Schrodinger equation on the cylinder. Our estimate is sharp, scale-invariant, and requires only L^2 data. Joint work with M. Christ and B. Pausader.

Zoom link. https://umn.zoom.us/j/95070384420

Title. Frobenius Theorem for Log-Lipschitz Subbundles

Abstract. In differential geometry, Frobenius theorem says that if a (smooth) real tangential subbundle is involutive, i.e. that X,Y are sections implies that [X,Y] is also a section, then this subbundle is spanned by some coordinate vector fields. Recently we prove the Frobenius theorem in the log-Lipschitz setting. In the talk I will go over the formulation of the theorem and show how harmonic analysis involves in the proof.

Zoom link. https://umn.zoom.us/j/92939569770

Title. Regular Lip(1,1/2) Approximation of Parabolic Hypersurfaces

Abstract. A classical result of David and Jerison states that a regular, n-dimensional set in R^{n+1} satisfying a two sided corkscrew condition is quantitatively approximated by Lipschitz graphs. After reviewing this result, we will discuss some recent advances in extending this result to the parabolic setting. The proofs of these results are quite difficult, but many of the underlying principles are easy to understand and quite geometric and presenting these geometric ideas will be the focus of this talk. As such, this talk will feature lots of pictures! Crucially, we highlight how fundamental differences of the parabolic setting require us to consider additional nuances which are not present in the elliptic setting. We will sketch the ideas of how to circumvent these difficulties.

Zoom link. https://umn.zoom.us/j/98551414461

Title. Effective equations of quantum mechanics and phase transitions

Abstract. Effective equations of many-body quantum mechanics form the backbone of many fields of modern physics. Notable examples of effective equations include the Hartree-Fock, Kohn-Sham, and Bogoliubov-de Gennes (BdG) equations. Although their physical derivations vary, we will review an unified formal mathematical frame work for their derivations (if time permits). In this frame work, the BdG equations are the most general form of effective equations. Physically, they form a microscopic description of superconductivity. When the temperature T is lower than a certain critical Tc, superconducting solutions emerge. In this talk, we will demonstrate the the existence of solutions to the BdG equations via variational arguments and show energy instability (hence the formation of a superconducting order parameter) when T < Tc.

This is a joint work with I. M. Sigal