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Methods Of Modern Mathematical Physics

Sets of Finite Perimeter and Geometric Variational Problems: Equilibrium shapes of liquids and sessile drops

Given a Hilbert space $${\mathcal{H}}$$
 , we investigate the well-posedness of the Cauchy problem for the wave equation for operators with a discrete non-negative spectrum acting on $${\mathcal{H}}$$
 . We consider the cases when the time-dependent propagation speed is regular, Holder, and distributional. We also consider cases when it is strictly positive (strictly hyperbolic case) and when it is non-negative (weakly hyperbolic case). When the propagation speed is a distribution, we introduce the notion of “very weak solutions” to the Cauchy problem. We show that the Cauchy problem for the wave equation with the distributional coefficient has a unique “very weak solution” in an appropriate sense, which coincides with classical or distributional solutions when the latter exist. Examples include the harmonic and anharmonic oscillators, the Landau Hamiltonian on $${\mathbb{R}^n}$$
 , uniformly elliptic operators of different orders on domains, Hormander’s sums of squares on compact Lie groups and compact manifolds, operators on manifolds with boundary, and many others.

/pdf/wave-equation-for-operators-with-discrete-spectrum-and-3l7sb5kz8k.pdf

Wave Equation for Operators with Discrete Spectrum and Irregular Propagation Speed

A distributional approach to fractional Sobolev spaces and fractional variation: Existence of blow-up

In this paper we study localization properties of the Riesz s-fractional gradient $$D^s u$$
 of a vectorial function u as $$s \nearrow 1$$
 . The natural space to work with s-fractional gradients is the Bessel space $$H^{s,p}$$
 for $$0< s < 1$$
 and $$1< p < \infty $$
 . This space converges, in a precise sense, to the Sobolev space $$W^{1,p}$$
 when $$s \nearrow 1$$
 . We prove that the s-fractional gradient $$D^s u$$
 of a function u in $$W^{1,p}$$
 converges strongly to the classical gradient Du. We also show a weak compactness result in $$W^{1,p}$$
 for sequences of functions $$u_s$$
 with bounded $$L^p$$
 norm of $$D^s u_s$$
 as $$s \nearrow 1$$
 . Moreover, the weak convergence of $$D^s u_s$$
 in $$L^p$$
 implies the weak continuity of its minors, which allows us to prove a semicontinuity result of polyconvex functionals involving s-fractional gradients defined in $$H^{s,p}$$
 to their local counterparts defined in $$W^{1,p}$$
 . The full $$\Gamma $$
 -convergence of the functionals is achieved only for the case $$p>n$$
 .

$$\Gamma $$-convergence of polyconvex functionals involving s-fractional gradients to their local counterparts

We continue the study of the space $BV^\alpha(\mathbb{R}^n)$ of functions with bounded fractional variation in $\mathbb{R}^n$ of order $\alpha\in(0,1)$ introduced in arXiv:1809.08575, by dealing with the asymptotic behaviour of the fractional operators involved. After some technical improvements of certain results of our previous work, we prove that the fractional $\alpha$-variation converges to the standard De Giorgi's variation both pointwise and in the $\Gamma$-limit sense as $\alpha\to1^-$. We also prove that the fractional $\beta$-variation converges to the fractional $\alpha$-variation both pointwise and in the $\Gamma$-limit sense as $\beta\to\alpha^-$ for any given $\alpha\in(0,1)$.

A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics II

We give sharp asymptotic estimates at infinity of all radial partial derivatives of the heat kernel on H-type groups. As an application, we give a new proof of the discreteness of the spectrum of some natural sub-Riemannian Ornstein–Uhlenbeck operators on these groups.

/pdf/asymptotics-for-the-heat-kernel-on-h-type-groups-42i8r8srpt.pdf

Asymptotics for the heat kernel on H-type groups

In this paper we study the boundedness of Bergman projectors on weighted Bergman spaces on homogeneous Siegel domains of Type II. As it appeared to be a natural approach in the special case of tube domains over irreducible symmetric cones, we study such boundedness on the scale of mixed-norm weighted Lebesgue spaces. The sharp range for the boundedness of such operators is essentially known only in the case of tube domains over Lorentz cones. In this paper we prove that the boundedness of such Bergman projectors is equivalent to variuos notions of atomic decomposition, duality, and characterization of boundary values of the mixed-norm weighted Bergman spaces, extending results moslty known only in the case of tube domains over irreducible symmetric cones. Some of our results are new even in the latter simpler context. We also study the simpler, but still quite interesting, case of the "positive" Bergman projectors, the integral operator in which the Bergman kernel is replaced by its absolute value. We provide a useful characterization which was previously known for tube domains. 

/pdf/boundedness-of-bergman-projectors-on-homogeneous-siegel-280gsghv.pdf

Boundedness of Bergman projectors on homogeneous Siegel domains

Given a graded group G and commuting, formally self-adjoint, left-invariant, homogeneous differential operators 
$$\mathcal {L}_1,\ldots , \mathcal {L}_n$$

 on G, one of which is Rockland, we study the convolution operators 
$$m(\mathcal {L}_1,\ldots , \mathcal {L}_n)$$

 and their convolution kernels, with particular reference to the case in which G is abelian and 
$$n=1$$

, and the case in which G is a 2-step stratified group which satisfies a slight strengthening of the Moore–Wolf condition and 
$$\mathcal {L}_1,\ldots ,\mathcal {L}_n$$

 are either sub-Laplacians or central elements of the Lie algebra of G. Under suitable conditions, we prove that (i) if the convolution kernel of the operator 
$$m(\mathcal {L}_1,\ldots , \mathcal {L}_n)$$

 belongs to 
$$L^1$$

, then m equals almost everywhere a continuous function vanishing at 
$$\infty $$

 (‘Riemann–Lebesgue lemma’); (ii) if the convolution kernel of the operator 
$$m(\mathcal {L}_1,\ldots , \mathcal {L}_n)$$

 is a Schwartz function, then m equals almost everywhere a Schwartz function.

Spectral Multipliers on 2-Step Stratified Groups, II

Given a 2-step stratified group which does not satisfy a slight strengthening of the Moore–Wolf condition, a sub-Laplacian $${\mathcal {L}}$$ and a family $${\mathcal {T}}$$ of elements of the derived algebra, we study the convolution kernels associated with the operators of the form $$m({\mathcal {L}}, -\,i {\mathcal {T}})$$. Under suitable conditions, we prove that: (i) if the convolution kernel of the operator $$m({\mathcal {L}},-\,i {\mathcal {T}})$$ belongs to $$L^1$$, then m equals almost everywhere a continuous function vanishing at $$\infty $$ (‘Riemann–Lebesgue lemma’); (ii) if the convolution kernel of the operator $$m({\mathcal {L}},-\,i{\mathcal {T}})$$ is a Schwartz function, then m equals almost everywhere a Schwartz function.

Mattia Calzi

Papers

A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics II

Asymptotics for the heat kernel on H-type groups

Boundedness of Bergman projectors on homogeneous Siegel domains

Spectral Multipliers on 2-Step Stratified Groups, II

Spectral Multipliers on 2-Step Stratified Groups, I