司靈得 (Daniel Spector)

I will speak about the limiting case of the Hardy--Littlewood--Sobolev inequality for \( p = 1 \). While the naive extension of HLS to \( p = 1 \) fails and the example that breaks the endpoint inequality is given by approximations of a delta measure, there are a few options how to obtain a correct inequality in the limit case. One of them, suggested by the work of Bourgain--Brezis, Van Schaftingen, and others, is to exclude the delta measures by imposing a linear translation and dilation invariant constraint on the functions in question. Another, suggested by Maz'ja, is based on adding certain non-linearity to the inequality. I will survey new results in this direction.

We discuss the solvability of Dirichlet problems of the type \(-\Delta_{p,w} u = \mu\) in \( \Omega \), \(u = 0\) on \( \partial \Omega \), where \( \Omega \) is a bounded domain in \( \mathbb{R}^{n} \), \( \Delta_{p,w} \) is a weighted (p, w)-Laplacian, and \( \mu \) is a nonnegative locally finite Radon measure on \( \Omega \). We do not assume the finiteness of \( \mu(\Omega) \). We revisit this problem from a potential theoretic perspective and provide criteria for the existence of solutions by \( L^{p}(w) \)- \( L^{p'}(\omega) \) trace inequalities or capacitary conditions. Additionally, we apply the method to the singular elliptic problem \( -\Delta_{p,w} u = \sigma|u|^{-s} \) in \( \Omega \), \(u = 0\) on \( \partial \Omega \), and derive connection with the trace inequalities.

The classical Sobolev embedding theorem says that the inequality \( \|f\|_{L^q(\mathbb{R}^d)} \lesssim \|\nabla f\|_{L^p(\mathbb{R}^d)}, \, f \in C_0^\infty(\mathbb{R}^d) \) 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 \( \|I_\alpha \, g\|_{L^q(\mathbb{R}^d)} \lesssim \|g\|_{L^p(\mathbb{R}^d)}, \, \frac{1}{p} - \frac{1}{q} = \frac{\alpha}{d}, \, 1 < p < q < \infty \), where \( I_\alpha \) 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.

The classical Sobolev embedding theorem says that the inequality \( \|f\|_{L^q(\mathbb{R}^d)} \lesssim \|\nabla f\|_{L^p(\mathbb{R}^d)}, \, f \in C_0^\infty(\mathbb{R}^d) \) 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 \( \|I_\alpha \, g\|_{L^q(\mathbb{R}^d)} \lesssim \|g\|_{L^p(\mathbb{R}^d)}, \, \frac{1}{p} - \frac{1}{q} = \frac{\alpha}{d}, \, 1 < p < q < \infty \), where \( I_\alpha \) 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.

Many problems from elasticity or fluid mechanics can be stated as \( \alpha \)-quasiconvex variational problems, a generalized 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.

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.