# FALL 2020 Nonlinear Analysis Seminar Series

### Abstract:

The boundedness properties of Calderón-Zygmund singular integral operators are of central importance in harmonic analysis, while the corresponding properties on weighted spaces has been of more recent interest. Indeed, within the last decade, optimal bounds for Calderón-Zygmund operators acting on weighted Lebesgue spaces have been obtained using sparse domination techniques. In addition to this theory concerning boundedness of Calderón-Zygmund operators, a theory for compactness of these operators has recently been established. In this talk, we present the extension of compact Calderón-Zygmund theory to weighted spaces using sparse domination methods. This work is joint with Paco Villarroya and Brett Wick.

### Abstract:

Classical works by F. Bethuel and by F. Hang and F-H. Lin have identified the local and global topological obstructions that prevent smooth maps from being dense in the Sobolev space $$W^{1, p}(M^{m}; N^{n})$$ between two Riemannian manifolds when $$p < m$$. They are related to the extension of continuous maps from subsets of $$M^{m}$$ to $$N^{n}$$. In this talk I will present some work in progress with P. Bousquet (Toulouse) and J. Van Schaftingen (UCLouvain), inspired from the notions of modulus introduced by B. Fuglede and degree for VMO maps by H. Brezis and L. Nirenberg. I shall explain how one can decide whether a specific Sobolev map $$u : M^{m} \to N^{n}$$ can be approximated or not by smooth ones, even in the presence of topological obstructions from $$M^{m}$$ or $$N^{n}$$.

### Abstract:

Plateau's problem in Euclidean space may be given many distinct formulations with solutions to most of them admitting an associated varifold. This includes Reifenberg's approach based on sets and Čech homology as well as Federer and Fleming's approach using integral currents and their homology. Thus, we employ the setting of varifolds to prove a priori bounds on the geodesic diameter in terms of boundary behaviour. This is ongoing joint work with C. Scharrer.

### Abstract:

The classical Sobolev embedding theorem says that the inequality $$\|f\|_{L_q(\mathbb{R}^d)}\lesssim \|\nabla f\|_{L_p(\mathbb{R}^d)},\quad f \in C_0^\infty(\mathbb{R}^d)$$ holds true provided $$\frac{1}{p} - \frac{1}{q} = \frac{1}{d}$$ and $$1 \leq p < d$$. The original Sobolev's proof was based on the Hardy--Littlewood--Sobolev (HLS) inequality $$\|\mathrm{I}_\alpha [g]\|_{L_q(\mathbb{R}^d)} \lesssim \|g\|_{L_p(\mathbb{R}^d)},\quad \frac{1}{p} - \frac{1}{q} = \frac{\alpha}{d},\ 1 < p < q < \infty,$$ here $$\mathrm{I}_{\alpha}$$ is the Riesz potential of order $$\alpha$$, i.e. a Fourier multiplier with the symbol $$|\cdot|^{-\alpha}$$. It is easy to see by plugging $$g = \delta_0$$ (the Dirac's delta) in the role of $$g$$ that the HLS inequality is false at the endpoint $$p=1$$. However, the Sobolev embedding is true in this case, as it was proved by Gagliardo and Nirenberg. The folklore principle, supported by the results of Bourgain--Brezis, Van Schaftingen, and many others, says that the HLS inequality becomes valid when we somehow separate the function $$g$$ from the set of delta-measures.  We will discuss this effect in more details and state new results in this direction.

### Abstract:

I will report about the theory of minimizing and critical knots under a set of scale invariant knot energies, the so-called tangent-point energy. We obtain lower semicontinuity and weak Sobolev-convergence of minimizing sequences to critical points away from finitely many points in the domain. Extending earlier work on Moebius-, and O'Hara energies we also obtain regularity for such critical points. This is based on joint work with S. Blatt, Ph. Reiter, and N. Vorderobermeier.

## ★SPECIAL LECTURE

### Abstract:

The classical Sobolev embedding theorem says that the inequality $$\|f\|_{L_q(\mathbb{R}^d)}\lesssim \|\nabla f\|_{L_p(\mathbb{R}^d)},\quad f \in C_0^\infty(\mathbb{R}^d)$$ holds true provided $$\frac{1}{p} - \frac{1}{q} = \frac{1}{d}$$ and $$1 \leq p < d$$. The original Sobolev's proof was based on the Hardy--Littlewood--Sobolev (HLS) inequality $$\|\mathrm{I}_\alpha [g]\|_{L_q(\mathbb{R}^d)} \lesssim \|g\|_{L_p(\mathbb{R}^d)},\quad \frac{1}{p} - \frac{1}{q} = \frac{\alpha}{d},\ 1 < p < q < \infty,$$ here $$\mathrm{I}_{\alpha}$$ is the Riesz potential of order $$\alpha$$, i.e. a Fourier multiplier with the symbol $$|\cdot|^{-\alpha}$$. It is easy to see by plugging $$g = \delta_0$$ (the Dirac's delta) in the role of $$g$$ that the HLS inequality is false at the endpoint $$p=1$$. However, the Sobolev embedding is true in this case, as it was proved by Gagliardo and Nirenberg. The folklore principle, supported by the results of Bourgain--Brezis, Van Schaftingen, and many others, says that the HLS inequality becomes valid when we somehow separate the function $$g$$ from the set of delta-measures. This talk is the second in a series of three in which we discuss in more detail the plan of the proofs.

### Abstract:

We discuss solvability of Dirichlet problems of the type $$- \Delta_{p} u = \mu$$ in $$\Omega$$, $$u = 0$$ on $$\partial \Omega$$, where $$\Omega$$ is a bounded domain, $$\Delta_{p}$$ is the p-Laplacian, and $$\mu$$ is a nonnegative locally finite Radon measure on $$\Omega$$. We do not assume the finiteness of $$\mu(\Omega)$$ here. We revisit this problem with a potential theoretic viewpoint and give sufficient conditions for the existence of solutions. Our main tools are $$L^{p}(dx)-L^{q}(d \mu)$$ trace inequalities and capacitary conditions. Also, we derive the trace inequalities using solutions conversely.

## ★SPECIAL LECTURE

### Abstract:

The classical Sobolev embedding theorem says that the inequality $$\|f\|_{L_q(\mathbb{R}^d)}\lesssim \|\nabla f\|_{L_p(\mathbb{R}^d)},\quad f \in C_0^\infty(\mathbb{R}^d)$$ holds true provided $$\frac{1}{p} - \frac{1}{q} = \frac{1}{d}$$ and $$1 \leq p < d$$. The original Sobolev's proof was based on the Hardy--Littlewood--Sobolev (HLS) inequality $$\|\mathrm{I}_\alpha [g]\|_{L_q(\mathbb{R}^d)} \lesssim \|g\|_{L_p(\mathbb{R}^d)},\quad \frac{1}{p} - \frac{1}{q} = \frac{\alpha}{d},\ 1 < p < q < \infty,$$ here $$\mathrm{I}_{\alpha}$$ is the Riesz potential of order $$\alpha$$, i.e. a Fourier multiplier with the symbol $$|\cdot|^{-\alpha}$$. It is easy to see by plugging $$g = \delta_0$$ (the Dirac's delta) in the role of $$g$$ that the HLS inequality is false at the endpoint $$p=1$$. However, the Sobolev embedding is true in this case, as it was proved by Gagliardo and Nirenberg. The folklore principle, supported by the results of Bourgain--Brezis, Van Schaftingen, and many others, says that the HLS inequality becomes valid when we somehow separate the function $$g$$ from the set of delta-measures. This talk is the third in a series of three in which we go into the proofs in more detail.

### Abstract:

In this talk we discuss how algebraic methods play a role in the sum-product problems in additive combinatorics. Moreover, we also discuss how sum-product estimates can be used to make non-trivial progress in geometric measure theory problems.

## ★SPECIAL LECTURE

### Abstract:

The classical Sobolev embedding theorem says that the inequality $$\|f\|_{L_q(\mathbb{R}^d)}\lesssim \|\nabla f\|_{L_p(\mathbb{R}^d)},\quad f \in C_0^\infty(\mathbb{R}^d)$$ holds true provided $$\frac{1}{p} - \frac{1}{q} = \frac{1}{d}$$ and $$1 \leq p < d$$. The original Sobolev's proof was based on the Hardy--Littlewood--Sobolev (HLS) inequality $$\|\mathrm{I}_\alpha [g]\|_{L_q(\mathbb{R}^d)} \lesssim \|g\|_{L_p(\mathbb{R}^d)},\quad \frac{1}{p} - \frac{1}{q} = \frac{\alpha}{d},\ 1 < p < q < \infty,$$ here $$\mathrm{I}_{\alpha}$$ is the Riesz potential of order $$\alpha$$, i.e. a Fourier multiplier with the symbol $$|\cdot|^{-\alpha}$$. It is easy to see by plugging $$g = \delta_0$$ (the Dirac's delta) in the role of $$g$$ that the HLS inequality is false at the endpoint $$p=1$$. However, the Sobolev embedding is true in this case, as it was proved by Gagliardo and Nirenberg. The folklore principle, supported by the results of Bourgain--Brezis, Van Schaftingen, and many others, says that the HLS inequality becomes valid when we somehow separate the function $$g$$ from the set of delta-measures.

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### Abstract:

Many problems from elasticity or fluid mechanics can be stated as $$\mathscr{A}$$-quasiconvex variational problems, a generalised variant of the usual notion of quasiconvexity due to Morrey. In this talk we give an overview of recent results on the (partial) regularity theory for such problems. As a main feature, we outline some novel links between harmonic analysis and the calculus of variations - comprising weighted singular integral estimates, coerciveness and regularity for variational integrals.

This talk comprises joint work with Sergio Conti.

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### Abstract:

In this talk,several classes of solutions will be treated for quasilinear elliptic equations of the type; $$-\Delta_p u = \sigma \, u^q +\mu$$ in $$\mathbb{R}^n$$ in the sub-natural growth case $$0 < q < p-1$$. Here $$\Delta_p$$ is the $$p$$-Laplacian, the coefficients $$\sigma$$ and data $$\mu$$ are nonnegative measurable functions (or measures). We will discuss pointwise estimates of solutions, as well as necessary and sufficient conditions for their existence.

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### Abstract:

I will discuss (L1,Lp) estimates for systems of PDEs of the form Au = 0, where A is a linear differential operator with constant coefficients and u is a vector-valued map satisfying a pointwise constraint of the form u(x) \in C, where C is a convex cone with sufficiently small aperture. I will collect some applications of this result to discuss higher integrability for Sobolev spaces and other spaces of bounded variation. This is joint work with G. De Philippis, J. Hirsch, F. Rindler and A. Skorobogatova.

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### Abstract:

We will discuss some recent results about commutators of certain Calderon-Zygmund operators and BMO spaces and how these generate bounded operators on Lebesgue spaces.  Results in other settings and other examples will be explained.  This talk is based on joint collaborative work.

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### Abstract:

We will review aspects of the theory of Compensated Compactness, starting with the fundamental work of Murat and Tartar and concluding with recent results obtained jointly with A. Guerra, J. Kristensen, and M. Schrecker. Broadly speaking, the object of this study is to gain a better understanding of the interaction between weakly convergent sequences and nonlinear functionals. The general framework will be that of variational integrals defined on spaces of vector fields satisfying linear pde constraints that satisfy Murat's constant rank condition. We will focus on the weak (lower semi-)continuity of these integrals, as well as the Hardy space regularity of the integrands.