We establish spectral estimates at a critical energy level for $h$-pseudors . Via a trace formula, we compute the contribution of isolated (non-extremum) critical points under a condition of "real principal type". The main result holds for all dimensions, for a singularity of any finite order and can be invariantly expressed in term of the geometry of the singularity. When the singularities are not integrable on the energy surface the results are significative since the order w.r.t. $h$ of the spectral distributions are bigger than in the regular setting.
We study the semi-classical trace formula at a critical energy level for an $h$-pseudo-differential operator on $\mathbb{R}^{n}$ whose principal symbol has a totally degenerate critical point for that energy. This problem is studied for a large time behavior and under the hypothesis that the principal symbol of the operator has a local extremum at the critical point.
We continue our study of a magnetic Schrödinger operator on a two-dimensional compact Riemannian manifold in the case when the minimal value of the module of the magnetic field is strictly positive. We analyze the case when the magnetic field has degenerate magnetic wells. The main result of the paper is an asymptotics of the groundstate energy of the operator in the semiclassical limit. The upper bounds are improved in the case when we have a localization by a miniwell effect of lowest order. These results are applied to prove the existence of an arbitrary large number of spectral gaps in the…
In this paper we establish a Lebeau-Robbiano spectral inequality for a degenerated one dimensional elliptic operator and show how it can be used to impulse control and finite time stabilization for a degenerated parabolic equation. R{é}sum{é} .-Dans cet article, on s'int{é}r{è}ss{è} a l'in{é}galit{é} spectrale de type Lebeau-Robbiano sur la somme de fonctions propres pour une famille d'op{é}rateurs d{é}g{é}n{é}r{é}s. Les applications sont donn{é}es en th{é}orie du contr{ô}le comme le contr{ô}le impulsionnel et la stabilisation en temps fini.
In this paper we study existence and spectral properties for weak solutions of Neumann and Dirichlet problems associated to second order linear degenerate elliptic partial differential operators $X$, with rough coefficients of the form $$X=-\text{div}(P\nabla )+{\bf HR}+{\bf S^\prime G} +F$$ in a geometric homogeneous space setting where the $n\times n$ matrix function $P=P(x)$ is allowed to degenerate. We give a maximum principle for weak solutions of $Xu\leq 0$ and follow this with a result describing a relationship between compact projection of the degenerate Sobolev space $QH^{1,p}$ into $…
For a degenerate hyperbolic equation of the second kind, and with a spectral parameter are studied the Cauchy problem, Cauchy-Goursat and Goursat in a new class of generalized solutions and is given an example that shows the importance of introducing the concept of such a class. Some operators with Bessel functions in the nucleus are introduced and their basic properties are studied. The important identities of these operators are helped to find an explicit representations of the stated problems.
We construct blowing-up sign-changing solutions to some nonlinear critical equations by glueing a standard bubble to a degenerate function. We develop a method based on analyticity to perform the glueing when the critical manifold of solutions is degenerate and no Bianchi--Egnell type condition holds.
This paper establishes the spectral stability in exponentially weighted spaces of smooth traveling monotone fronts for reaction diffusion equations of Fisher-KPP type with nonlinear degenerate diffusion coefficient. It is assumed that the former is degenerate, that is, it vanishes at zero, which is one of the equilibrium points of the reaction. A parabolic regularization technique is introduced in order to locate a subset of the compression spectrum of the linearized operator around the wave, whereas the point spectrum is proved to be stable with the use of energy estimates. Detailed asymptoti…
In this paper we prove a Morse Lemma for degenerate critical points of a function u which satisfies -Δu=f(u) in B_1, where B_1 is the unit ball of R^2 and f is a smooth nonlinearity. Other results on the nondegeneracy of the critical points and the shape of the level sets are proved.
We study the semi-classical trace formula at a critical energy level for a Schrödinger operator on $\mathbb{R}^{n}$. We assume here that the potential has a totally degenerate critical point associated to a local maximum. The main result, which establishes the contribution of the associated equilibrium in the trace formula, is valid for all time in a compact subset of $\mathbb{R}$ and includes the singularity in $t=0$. For these new contributions the asymptotic expansion involves the logarithm of the parameter $h$. Depending on an explicit arithmetic condition on the dimension and the order of…
This paper is devoted to the study of degenerate critical elliptic equations of Caffarelli-Kohn-Nirenberg type. By means of blow-up analysis techniques, we prove an a-priori estimate in a weighted space of continuous functions. From this compactness result, the existence of a solution to our problem is proved by exploiting the homotopy invariance of the Leray-Schauder degree.
The aim of this paper is to offer an original and comprehensive spectral theoretical approach to the study of convergence to equilibrium, and in particular of the hypocoercivity phenomenon, for contraction semigroups in Hilbert spaces. Our approach rests on a commutation relationship for linear operators known as intertwining, and we utilize this identity to transfer spectral information from a known, reference semigroup $\tilde{P} = (e^{-t\tilde{\mathbf A}})_{t \geq 0}$ to a target semigroup $P$ which is the object of study. This allows us to obtain conditions under which $P$ satisfies a hypo…
We prove the local Lipschitz continuity and the higher differentiability of local minimizers of integral functionals with non autonomous integrand which is degenerate convex with respect to the gradient variable. The main novelty here is that the results are obtained assuming that the coefficients have weak derivative in an almost critical Zygmund class and the datum f is assumed to belong to the same Zygmund class.
This paper is devoted to the analysis of non-negative solutions for a degenerate parabolic-elliptic Patlak-Keller-Segel system with critical nonlinear diffusion in a bounded domain with homogeneous Neumann boundary conditions. Our aim is to prove the existence of a global weak solution under a smallness condition on the mass of the initial data, there by completing previous results on nite blow-up for large masses. Under some higher regularity condition on solutions, the uniqueness of solutions is proved by using a classical duality technique.
We prove the existence of a critical Fujita exponent for a non-homogeneous semilinear heat equation which involves degenerate coefficients. More precisely, in order to give a rather complete theory, we focus on two types of weights $w(x)=|x_1|^a$ or $w(x)=|x|^b$ where $a, b>0$ in a suitable range. The coefficients under consideration admit either a singularity at the origin or a line of singularities. In the latter case, the problem is related to the fractional Laplacian.
In this paper, we study the following degenerate critical elliptic equations with anisotropic coefficients $$ -div(|x_{N}|^{2α}\nabla u)=K(x)|x_{N}|^{α\cdot 2^{*}(s)-s}|u|^{2^{*}(s)-2}u {in} \mathbb{R}^{N} $$ where $x=(x_{1},...,x_{N})\in\mathbb{R}^{N},$ $N\geq 3,$ $α>1/2,$ $0\leq s\leq 2$ and $2^{*}(s)=2(N-s)/(N-2).$ Some basic properties of the degenerate elliptic operator $-div(|x_{N}|^{2α}\nabla u)$ are investigated and some regularity, symmetry and uniqueness results for entire solutions of this equation are obtained. We also get some variational identities for solutions of this equation.…
We investigate a non-homogeneous nonlinear heat equation which involves degenerate or singular coefficients belonging to the $A_2$ class of functions. We prove the existence of a Fujita exponent and describe the dichotomy existence/non-existence of global in time solutions. The $A_2$ coefficient admits either a singularity at the origin or a line of singularities. In this latter case, the problem is related to the fractional laplacian, through the Caffarelli-Silvestre extension and is a first attempt to develop a parabolic theory in this setting.
In the paper, we study the Prandtl system with initial data admitting non-degenerate critical points. For any index $σ\in[3/2, 2],$ we obtain the local in time well-posedness in the space of Gevrey class $G^σ$ in the tangential variable and Sobolev class in the normal variable so that the monotonicity condition on the tangential velocity is not needed to overcome the loss of tangential derivative. This answers the open question raised in the paper of D. Gérard-Varet and N. Masmoudi [{\it Ann. Sci. Éc. Norm. Supér}. (4) 48 (2015), no. 6, 1273-1325], in which the case $σ=7/4$ is solved.
We study the minimizer u of a convex functional in the plane which is not Gâteaux-differentiable. Namely, we show that the set of critical points of any C^1-smooth minimizer can not have isolated points. Also, by means of some appropriate approximating scheme and viscosity solutions, we determine an Euler-Lagrange equation that u must satisfy. By applying the same approximating scheme, we can pair u with a function v which may be regarded as the stream function of u in a suitable generalized sense.
The aim of this paper is to study global bifurcations of non-constant solutions of some nonlinear elliptic systems, namely the system on a sphere and the Neumann problem on a ball. We study the bifurcation phenomenon from families of constant solutions given by critical points of the potentials. Considering this problem in the presence of additional symmetries of a compact Lie group, we study orbits of solutions and, in particular, we do not require the critical points to be isolated. Moreover, we allow the considered orbits of critical points to be degenerate. To prove the bifurcation we comp…
We consider degenerate differential operators $A = \displaystyle{\sum_{k,j=1}^d \partial_k (a_{kj} \partial_j)}$ on $L^2(\mathbb{R}^d)$ with real symmetric bounded measurable coefficients. Given a function $χ\in C_b^\infty(\mathbb{R}^d)$ (respectively, $Ω$ a bounded Lipschitz domain) and suppose that $(a_{kj}) \ge μ> 0$ a.e.\ on $ \supp χ$ (resp., a.e.\ on $Ω$). We prove a spectral multiplier type result: if $F\colon [0, \infty) \to \mathbb{C}$ is such that $\sup_{t > 0} \| φ(.) F(t .) \|_{C^s} < \infty$ for some non-trivial function $φ\in C_c^\infty(0,\infty)$ and some $s > d/2$ then $M_χF(I+…
The mixed problem for a degenerate high order equation with a fractional derivative in a rectangular domain is considered in the article. The existence of a solution and its uniqueness are shown by the spectral method.
In this paper, we are concerned with semiclassical states to the following Sobolev critical Dirac equation with degenerate potential, \begin{align*} -\textnormal{i} \eps α\cdot \nabla u + a βu + V(x) u=|u|^{q-2} u + |u| u \quad \mbox{in} \,\, \R^3, \end{align*} where $u:\mathbb{R}^3\rightarrow \mathbb{C}^4$, $2<q<3$, $\eps>0$ is a small parameter, $a>0$ is a constant, $α=(α_1, α_2, α_3)$, $α_j$ and $β$ are $4 \times 4$ Pauli-Dirac matrices. We construct an infinite sequence of higher topological type semiclassical states with higher energies concentrating around the local minimum points of the…
We are concerned with a class of degenerate diffusion equations with time delay describing population dynamics with age structure. In our recent study [{\em Nonlinearity}, 33 (2020), 4013--4029], we established the existence and uniqueness of critical traveling wave for the time-delayed degenerate diffusion equations, and obtained the reducing mechanism of time delay on critical wave speed. In this paper, we now are able to show the asymptotic spreading speed and its coincidence with the critical wave speed $c^*(m,r)$ of sharp wave, and prove that the initial perturbation or the boundary of th…
We study the semi-classical trace formula at a critical energy level for a $h$-pseudo-differential operator whose principal symbol has a unique non-degenerate critical point for that energy. This leads to the study of Hamiltonian systems near equilibrium and near the non-zero periods of the linearized flow. The contributions of these periods to the trace formula are expressed in terms of degenerate oscillatory integrals. The new results obtained are formulated in terms of the geometry of the energy surface and the classical dynamics on this surface.
This paper is concerned with the critical sharp traveling wave for doubly nonlinear diffusion equation with time delay, where the doubly nonlinear degenerate diffusion is defined by $\Big(\big|(u^m)_x\big|^{p-2}(u^m)_x\Big)_x$ with $m>0$ and $p>1$. The doubly nonlinear diffusion equation is proved to admit a unique sharp type traveling wave for the degenerate case $m(p-1)>1$, the so-called slow-diffusion case. This sharp traveling wave associated with the minimal wave speed $c^*(m,p,r)$ is monotonically increasing, where the minimal wave speed satisfies $c^*(m,p,r)<c^*(m,p,0)$ for any time del…
We consider the second order Cauchy problem $$u''+m(|A^{1/2}u|^2)Au=0, u(0)=u_{0}, u'(0)=u_{1},$$ where $m:[0,+\infty)\to[0,+\infty)$ is a continuous function, and $A$ is a self-adjoint nonnegative operator with dense domain on a Hilbert space. It is well known that this problem admits local-in-time solutions provided that $u_{0}$ and $u_{1}$ are regular enough, depending on the continuity modulus of $m$, and on the strict/weak hyperbolicity of the equation. We prove that for such initial data $(u_{0},u_{1})$ there exist two pairs of initial data $(\overline{u}_{0},\overline{u}_{1})$, $(\wideh…
We study critical and stationary, i.e. critical with respect to both inner and outer variations, points of polyconvex functionals of the form $f(X) = g(\det(X))$, for $X \in \mathbb{R}^{2\times 2}$. In particular, we show that critical points $u \in Lip(Ω,\mathbb{R}^2)$ with $\det(Du) \neq 0$ a.e. have locally constant determinant except in a relatively closed set of measure zero, and that stationary points have constant determinant almost everywhere. This is deduced from a more general result concerning solutions $u \in Lip(Ω,\mathbb{R}^n)$, $Ω\subset \mathbb{R}^n$ to the linearized problem $…
We study the spectral structure of the complex linearized operator for a class of nonlinear Schrödinger systems, obtaining as byproduct some interesting properties of non-degenerate ground state of the associated elliptic system, such as being isolated and orbitally stable.
Well-posedness of the two-dimensional unsteady Prandtl system in Sobolev space with degenerate critical points
College of Mathematics and Systems Science, Shandong University of Science and Technology（山东科技大学数学与系统科学学院）；School of Mathematical Sciences, MOE-LSC and SHL-MAC, Shanghai Jiao Tong University（上海交通大学数学科学学院，教育部低维材料物理化学重点实验室和上海市应用数学实验室）
This paper deals with the following degenerate fractional Kirchhoff-type system with magnetic fields and critical growth: $$ \left\{ \begin{array}{lll} -\mathfrak{M}(\|u\|_{s,A}^2)[(-Δ)^s_Au+u] = G_u(|x|,|u|^2,|v|^2) + \left(\mathcal{I}_μ*|u|^{p^*}\right)|u|^{p^*-2}u \ &\mbox{in}\,\,\mathbb{R}^N,\\ \mathfrak{M}(\|v\|_{s,A})[(-Δ)^s_Av+v] = G_v(|x|,|u|^2,|v|^2) + \left(\mathcal{I}_μ*|v|^{p^*}\right)|v|^{p^*-2}v \ &\mbox{in}\,\,\mathbb{R}^N, \end{array}\right. $$ where $$\|u\|_{s,A}=\left(\iint_{\mathbb{R}^{2N}}\frac{|u(x)-e^{i(x-y)\cdot A(\frac{x+y}{2})}u(y)|^2}{|x-y|^{N+2s}}dx dy+\int_{\mathbb{…
This paper is devoted to the analysis of non-negative solutions for a generalisation of the classical parabolic-elliptic Patlak-Keller-Segel system with $d\ge3$ and porous medium-like non-linear diffusion. Here, the non-linear diffusion is chosen in such a way that its scaling and the one of the Poisson term coincide. We exhibit that the qualitative behaviour of solutions is decided by the initial mass of the system. Actually, there is a sharp critical mass $M_c$ such that if $M \in (0,M_c]$ solutions exist globally in time, whereas there are blowing-up solutions otherwise. We also show the ex…
Let $(M,g)$ be a 3-dimensional Riemannian manifold. The goal of the paper it to show that if $P_{0}\in M$ is a non-degenerate critical point of the scalar curvature, then a neighborhood of $P_{0}$ is foliated by area-constrained Willmore spheres. Such a foliation is unique among foliations by area-constrained Willmore spheres having Willmore energy less than $32π$, moreover it is regular in the sense that a suitable rescaling smoothly converges to a round sphere in the Euclidean three-dimensional space. We also establish generic multiplicity of foliations and the first multiplicity result for …
In this article we introduce and investigate a new two-parameter family of knot energies $TP^{(p,q)}$ that contains the tangent-point energies. These energies are obtained by decoupling the exponents in the numerator and denominator of the integrand in the original definition of the tangent-point energies. We will first characterize the curves of finite energy in the sub-critical range $p\in(q+2,2q+1)$ and see that those are all injective and regular curves in the Sobolev-Slobodeckiĭ space $W^{(p-1)/q,q}$. We derive a formula for the first variation that turns out to be a non-degenerate ellipt…
We prove a new, universal gradient continuity estimate for solutions to quasilinear equations with varying coefficients at points on its critical singular set of degeneracy $S(u) := \{X : D u(X) = 0 \}$. Our main Theorem reveals that along $S(u)$, $u$ is asymptotically as regular as solutions to constant coefficient equations. In particular, along the critical set $S(u)$, $Du$ enjoys a modulus of continuity much superior than the, possibly low, continuity feature of the coefficients. Our main, leading result fosters a new understanding on smoothness properties of solutions to degenerate or sin…
We consider a degenerate chemotaxis model with two-species and two-stimuli in dimension $d\geq 3$ and find two critical curves intersecting at one same point which separate the global existence and blow up of weak solutions to the problem. More precisely, above these curves (i.e. subcritical case), the problem admits a global weak solution obtained by the limits of strong solutions to an approximated system. Based on the second moment of solutions, initial data are constructed to make sure blow up occurs in finite time below these curves (i.e. critical and supercritical cases). In addition, th…
In this paper, we obtain new Carleman estimates for a class of variable coefficient degenerate elliptic operators whose constant coefficient model at one point is the so called Baouendi-Grushin operator. This generalizes the results obtained by the two of us with Garofalo in [9] where similar estimates were established for the "constant coefficient" Baouendi-Grushin operator. Consequently, we obtain: (i) a Bourgain-Kenig type quantitative uniqueness result in the variable coefficient setting; (ii) and a strong unique continuation property for a class of degenerate sublinear equations. We also …
We establish sharp long time asymptotic behaviour for a family of entropies to defective Fokker-Planck equations and show that, much like defective finite dimensional ODEs, their decay rate is an exponential multiplied by a polynomial in time. The novelty of our study lies in the amalgamation of spectral theory and a quantitative non-symmetric hypercontractivity result, as opposed to the usual approach of the entropy method.
In this paper, we study the separability and spectral properties of singular degenerate elliptic equations in vector valued spaces. We prove that a realization operator by this equation with some boundary conditions is separable and Fredholm in Lebesque spaces. The leading part of the associated differential operator is not self-adjoint. The sharpe estimate of the resolvent, discreetness of spectrum and completeness of root elements of this operator is obtained
We study a one-dimensional parabolic PDE with degenerate diffusion and non-Lipschitz nonlinearity involving the derivative. This evolution equation arises when searching radially symmetric solutions of a chemotaxis model of Patlak-Keller-Segel type. We prove its local in time wellposedness in some appropriate space, a blow-up alternative, regularity results and give an idea of the shape of solutions. A transformed and an approximate problem naturally appear in the way of the proof and are also crucial in [22] in order to study the global behaviour of solutions of the equation for a critical pa…
