The structure of the singular set in the thin obstacle problem for degenerate parabolic equations

We study the singular set in the thin obstacle problem for degenerate parabolic equations with weight $$|y|^a$$ | y | a for $$a \in (-1,1)$$ a ∈ ( - 1 , 1 ) . Such problem arises as the local extension of the obstacle problem for the fractional heat operator $$(\partial _t - \Delta _x)^s$$ ( ∂ t - Δ x ) s for $$s \in (0,1)$$ s ∈ ( 0 , 1 ) . Our main result establishes the complete structure and regularity of the singular set of the free boundary. To achieve it, we prove Almgren-Poon, Weiss, and Monneau type monotonicity formulas which generalize those for the case of the heat equation ($$a=0$$ a = 0 ).

SOLVABILITY OF A NON–LOCAL PROBLEM FOR A THIRD—ORDER EQUATION WITH THE HEAT OPERATOR IN THE MAIN PAR

В работе доказаны теоремы существования и единственности регулярных решений одной нелокальной задаче с интегральным условием для уравнения третьего порядка с оператором теплопроводности в главной части. Доказательство основано на сведение поставленной задачи к смешанной задаче для нагруженного уравнения теплопроводности. In this paper, we considered the solvability of a nonlocal problem with integral condition for a thirdorder equation with head operatot in the main part. The existence and uniqueness of a regular solution to this problem are proved. The proof is based on reducing a non-local problem to the mixed problem for a loaded heat equation

Generalized Wehrl entropies and Euclidean Landau levels

We are concerned with a class of generalized coherent states (GCS) attached to Euclidean Landau levels (or to [Formula: see text]-true-polyanalytic spaces), which can be obtained by displacing a Gaussian–Hermite function as an admissible (or window) function. Precisely, we evaluate the Wehrl entropy for a density operator representing a projector on a Fock state (pure states) and we give an upper bound for this entropy. We also establish an exact formula of this entropy for the heat operator (mixed states) associated with the harmonic oscillator. In this case, the behavior of the entropy with respect to the temperature parameter shows the dependence of its minimum to the Landau level or equivalently to the window function by means of which the GCS involved in the Wehrl entropy were constructed.

Commutators of the fractional integrals for second-order elliptic operators on Morrey spaces

Let {L=-\operatorname{div}(A\nabla)} be a second-order divergence form elliptic operator and let A be an accretive, {n\times n} matrix with bounded measurable complex coefficients in {{\mathbb{R}}^{n}}. Let {L^{-\frac{\alpha}{2}}} be the fractional integral associated to L for {0<\alpha<n}. For {b\in L_{\mathrm{loc}}({\mathbb{R}}^{n})} and {k\in{\mathbb{N}}}, the k-th order commutator of b and {L^{-\frac{\alpha}{2}}} is given by(L^{-\frac{\alpha}{2}})_{b,k}f(x)=L^{-\frac{\alpha}{2}}((b(x)-b)^{k}f)(x).In the paper, we mainly show that if {b\in\mathrm{BMO}({\mathbb{R}}^{n})}, {0<\lambda<n} and {0<\alpha<n-\lambda}, then {(L^{-\frac{\alpha}{2}})_{b,k}} is bounded from {L^{p,\lambda}} to {L^{q,\lambda}} for {p_{-}(L)<p<q<p_{+}(L)\frac{n-\lambda}{n}} and {\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n-\lambda}}, where {p_{-}(L)} and {p_{+}(L)} are the two critical exponents for the {L^{p}} uniform boundedness of the semigroup {\{e^{-tL}\}_{t>0}}. Also, we establish the boundedness of the commutator of the fractional integral with Lipschitz function on Morrey spaces. The results encompass what is known for the classical Riesz potentials and elliptic operators with Gaussian domination by the classical heat operator.

Isoperimetric inequalities for the Cauchy-Dirichlet heat operator

In this paper we prove that the first s-number of the Cauchy-Dirichlet heat operator is minimized in a circular cylinder among all Euclidean cylindric domains of a given measure. It is an analogue of the Rayleigh-Faber-Krahn inequality for the heat operator. We also prove a Hong-Krahn-Szeg? and a Payne-P?lya-Weinberger type inequalities for the Cauchy-Dirichlet heat operator.