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Current contact: Gerardo Mendoza
The seminar takes place Mondays 2:30 - 3:30 pm either in Wachman 617 or virtually via Zoom. Please see the announcment of the specific meeting. For virtual meetings, please write if you wish to join. Click on title for abstract.
We prove sufficient conditions for a Calder\'on-Zygmund operator to belong to the $p$-th Schatten-von Neumann classes $S_{p}(L^{2}(\mathbb R^{d}))$.
As in the classical $T1$ theory, the conditions are given in terms of the smoothness of the operator kernel, and the action of both the operator and its adjoint on the function $1$.
To prove membership to the Schatten class when $p>2$ we develop new bump estimates for composed compact Calder\'on-Zygmund operators, and a new extension of Carleson's Embedding Theorem.
Linear PDOs can act on various (generalized) function spaces, provided their coefficients are sufficiently regular: smooth, real-analytic and/or Gevrey spaces, as well as their generalized counterparts, just to name a few. It may then be of interest to establish properties of regularity and solvability (either local or global; several flavors of hypoellipticity; properties of the associated cohomology spaces for systems; and so on) of such PDOs in some of these spaces, sometimes providing radically different answers depending on the space under study.
Some results in the literature, however, establish relationships among such properties for whole classes of operators:
-- Greenfield (1972) proved that for operators with constant coefficients on tori global hypoellipticity implies global analytic-hypoellipticity.
-- Dealing with differential complexes associated to locally integrable structures, Caetano and Cordaro (2011) proved that if in a given degree the complex is locally solvable in the smooth setup then it is also locally solvable in the Gevrey setup (same degree), while Ragognette (2019), using similar methods, relates these with local solvability in the sense of Gevrey ultradistributions.
-- Still dealing with locally integrable structures, Malaspina and Nicola (2014) conjecture another connection between smooth and Gevrey local solvability (a kind of converse to the result of Caetano and Cordaro), which is currently open except for a few cases.
-- In a joint work with Cordaro (2019) on analytic structures we connect (in a few particular cases) local solvability in the smooth sense with a property called semi-local analytic solvability.
Here we are interested in global properties for systems of left-invariant differential operators on compact Lie groups: regularity properties, properties on the closedness of the range and dimension of cohomology spaces for complexes, when acting on various function spaces. Extending the methods of Greenfield and Wallach (1973) to systems, we obtain abstract characterizations for these properties and use them to derive some generalizations of Greenfield's result, as well as global versions of the result of Caetano and Cordaro for left-invariant involutive structures.
For a nonsingular planar vector field $L$ with complex-valued coefficient that has local first integrals that are open maps, we consider the equation $Lu=f$ and show its solutions can be represented through a generalized Cauchy integral operator.
In this talk I will first talk about the isospectral problem in geometry and about isospectrality of Strum-Liouville operators on a finite interval in the simplified form of a Schrödinger operator. I will mention very interesting known results about isospectral potentials. I will introduce generalizations of the concept of isospectrality like quasi-isospectrality, and will present what we know so far about quasi-isospectral potentials. The work presented here is still on-going joint work with Camilo Perez.
In this talk we will consider the essential spectrum of the Laplacian on differential forms over noncompact manifolds. We will see a brief overview of known results and discuss the main differences between the function and form spectrum. One interesting problem in the area is finding sufficient and general enough conditions on the manifold so that the essential spectrum on forms is a connected set. We will see that over asymptotically flat manifolds this is the case. The proof involves the study of the structure of the manifold at infinity via Cheeger-Fukaya-Gromov theory and Cheeger-Colding theory, combined with a generalized Weyl criterion for the computation of the spectrum. Finally, we present some recent results on the form spectrum of negatively curved manifolds.
This will be a talk at in harmonic analysis with little bits of number theory. We will discuss some of the different faces of the uncertainty principle for the Fourier transform and its recent connections to lattices and packing problems, and then slowly move towards uncharted territories. The required background in analysis will be minimal.
I will present a result on Sobolev regularity of weak solutions to linear nonlocal equations. The theory we develop is concerned with obtaining higher integrability and differentiability of solutions of nonlocal equations. Under the assumption of uniform Holder continuity of coefficients, weak solutions from the energy space that correspond to highly integrable right hand side will be shown to have improved Sobolev regularity along the differentiability scale in addition to the expected integrability gain. This result is consistent with self-improving properties of nonlocal equations that has been observed by other earlier works. To prove our result, we use a perturbation argument where optimal regularity of solutions of a simpler equation is systematically used to derive an improved regularity for the solution of the nonlocal equation.
No meeting
In collaboration with Gabriel Araújo (ICMC-USP) and Igor A. Ferra (Federal Univ. of ABC), we studied global solvability of operators on compact manifolds.
The goals of this talk are to discuss how a weak notion of global hypoellipticity implies global solvability and also to give necessary and sufficient conditions for global solvability of a class of operators of type sum of squares defined on a product compact manifolds.
It is well known that the De Rham cohomology of a compact Lie group is isomorphic to the Chevalley-Eilenberg complex. While the former is a topological invariant of the Lie group, the latter can be computed by using simple linear algebra methods. In this talk, we discuss how to obtain an injective homomorphism between the cohomology spaces associated with left-invariant involutive structures and the cohomology of a generalized Chevalley-Eilenberg complex.
We discuss some cases in which the homomorphism is surjective, such as the Dolbeault cohomology and certain elliptic and CR structures. The results provide new insights regarding the general theory of involutive structures as, for example, they reveal algebraic obstructions for solvability for the associated differential complexes.
This research originates from recent results by M. Goldman and F. Otto concerning regularity of optimal transport maps for the quadratic cost. We consider cost functions having the form $c(x,y)=h(x-y)$, where $h$ is positively homogeneous of degree $p>1$ and $h\in C^2(\mathbb R^n\setminus \{0\})$. A mapping $T:\mathbb R^n\to \mathbb R^n$ is $c$-monotone if $c(Tx,x)+c(Ty,y)\leq c(Tx,y)+c(Ty,x)$. Using Green's representation formulas, if $T$ is $c$-monotone, we prove local $L^\infty$-estimates of $Tx-x$ in terms of $L^p$-averages of $Tx-x$. From this we deduce estimates for the interpolating maps between $T$ and $Id$, and when $T$ is optimal, $L^\infty$-estimates of $T^{-1}x-x$. As a consequence of the technique, we also obtain a.e. differentiability of monotone maps.
This is joint work with Annamaria Montanari (Bologna) to appear in Calculus of Variations and PDEs.
The trademark blueprint of a Fatou-type theorem is that size/ntegrability properties of the nontangential maximal operator for a null-solution of an elliptic equation in a certain domain implies the a.e. existence of the pointwise nontangential boundary trace of the said function. It is natural to call such a theorem quantitative if the boundary trace does not just simply exists but encodes significant information regarding the size of the original function.
In this talk, which is based on joint work with Dorina Mitrea (Baylor) and Irina Mitrea (Temple), I will be presenting a quantitative Fatou-type theorem for null-solutions of an injectively elliptic first-order (homogeneous, constant complex coefficient) system of differential operators in an arbitrary uniformly rectifiable domain in the $n$-dimensional Euclidean space, assuming that the nontangential maximal operator is $p$-th power integrable (with respect to the Hausdorff measure) for some integrability exponent larger than $(n-1)/n$. Such a result has a wide range of applications, including the theory of Hardy spaces associated with injectively elliptic first-order systems in uniformly rectifiable domains.
Compensated compactness is an amazing result, originally due to Murat and Tartar, that states that the dot product of two weakly convergent in $L^2$ sequences of vector fields converges to the dot product of their weak limits, provided one of the sequences is curl-free, and the other is divergence-free. I will show how to generalize this result to a much larger class of differential operators and then use it to prove a homogenization theorem for a large class of elliptic systems of PDEs.
This talk is specifically aimed at graduate students, especially the ones taking Functional Analysis.
The subject of this talk is the analysis of multilayer potentials associated with integer powers of the classical $\overline{\partial}$ operator in non-smooth domains in the complex plane. This analysis includes integral representation formulas, jump relations, higher-order Fatou theorems, and higher-order Hardy spaces.
This is joint work with Irina Mitrea (Temple University), Dorina Mitrea and Marius Mitrea (Baylor University).
The inequalities of Petty and Zhang are affine isoperimetric inequalities, the former of which implies that classical isoperimetric inequality and is equivalent to an affine version of the Sobolev inequality for compactly support $C^1$ functions, while the latter is a very strong reverse isoperimetric inequality. Each of these inequalities feature a certain class of convex bodies, called projection bodies, which may be described in terms of the cosine transform of the surface area measure of a given convex body.
In this talk, we will discuss a generalization of these bodies to the weighted setting (by replacing the surface area measure with different measures satisfying mild regularity conditions) and describe how they may be used to prove strong reverse isoperimetric inequalities. And, in addition, show how these results may be used to imply a reverse form of the isoperimetric inequality for certain classes of measures on the $n$-dimensional Euclidean space (the Lebesgue measure and Gaussian measure, for example).
This is based on a joint work with D. Langharst and A. Zvavitch.
We consider an electroconvection model describing the evolution of a charge density carried by a two-dimensional incompressible fluid flowing through a porous medium. Electrical forces are created by the charge density and balanced by Darcy's law. The resulting partial differential equation obeyed by the charge density is nonlinear and nonlocal. In this talk, we study the global existence, uniqueness, and regularity of solutions to the model for small initial data.
The Boussinesq equations are a member of a family of models of incompressible fluid equations, including the 3D Euler equations, for which the problem of global existence of solutions is open. The Boussinesq equations arise in fluid mechanics, in connection to thermal convection and they are extensively studied in that context. Formation of finite time singularities from smooth initial data in ideal (conservative) 2D Boussinesq equations is an important open problem, related to the blow up of solutions in 3D Euler equations. The Voigt Boussinesq is a conservative approximation of the Boussinesq equations which has certain attractive features, including sharing the same steady solutions with the Boussinesq equations. In this talk, after giving a brief description of issues of local and global existence, well-posedness and approximation in the incompressible fluids equations, I will present a global regularity result for critical Voigt Boussinesq equations.
Some mathematical formulations of physical phenomena correspond to overdetermined boundary value problems, that is, boundary problems in which one prescribes both Dirichlet and Neumann type boundary datum.
The subject of this talk is the analysis of overdetermined boundary value problems (OBVP) for the Laplacian in non-smooth domains with boundary datum in Whitney--Lebesgue spaces with integrability index in the interval $(1,\infty)$. This analysis includes integral representation formula, jump relations, and solvability of the OBVP in uniformly rectifiable domain.
This is joint work with Irina Mitrea (Temple University), Dorina Mitrea and Marius Mitrea (Baylor University).
Conservation laws and Lyapunov functions are powerful tools for proving the global existence of stability of solutions, but for many complex systems these tools are insufficient to completely understand non-perturbative dynamics. In this talk I will discuss a complex-scalar PDE which may be seen as a toy model for vortex stretching in fluid flow, and cannot be neatly categorized as conservative nor dissipative.
In a recent series of papers we have shown that this equation exhibits rich dynamical behavior that exist globally in time: non-trivial equilibria, homoclinic orbits, heteroclinic orbits, and integrable subsystems foliated by periodic orbits. On the other side of the coin, we show several mechanisms by which solutions can blowup. I will discuss these results, and current work toward understanding unstable blowup.
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