In this paper, we establish a fundamental inequality for fourth order partial differential operator $\cal P=α\partial_s+β\partial_{ss}+Δ^2$ ($α, β\in\mathbb{R}$) with an abstract exponential-type weight function. Such kind of weight functions including not only the regular weight functions but also the singular weight functions. Using this inequality we are able to prove some Carleman estimates for the operator $\cal P$ with some suitable boundary conditions in the case of $β<0$ or $α\neq 0, β=0$. As application, we obtain a resolvent estimate for $\cal P$, which can imply a log-type stabilization result for the plate equation with clamped boundary conditions or hinged boundary conditions.
New embeddings of weighted Sobolev spaces are established. Using such embeddings, we obtain the existence and regularity of positive solutions with Navier boundary value problems for a weighted fourth order elliptic equation. We also obtain Liouville type results for the related equation. Some problems are still open.
We consider the time local and global well-posedness for the fourth order nonlinear Schrodinger type equation (4NLS) on the torus. The nonlinear term of (4NLS) contains the derivatives of unknown function and this prevents us to apply the classical energy method. To overcome this difficulty, we introduce the modified energy and derive an a priori estimate for the solution to (4NLS).
We prove that the following pointwise inequality holds \begin{equation*} -Δu \ge \sqrt\frac{2}{(p+1)-c_n} |x|^{\frac{a}{2}} u^{\frac{p+1}{2}} + \frac{2}{n-4} \frac{|\nabla u|^2}{u} \ \ \text{in}\ \ \mathbb{R}^n \end{equation*} where $c_n:=\frac{8}{n(n-4)}$, for positive bounded solutions of the fourth order Hénon equation that is \begin{equation*} Δ^2 u = |x|^a u^p \ \ \ \ \text {in }\ \ \mathbb{R}^n \end{equation*} for some $a\ge0$ and $p>1$. Motivated by the Moser's proof of the Harnack's inequality as well as Moser iteration type arguments in the regularity theory, we develop an iteration a…
We deal with nonlinear weighted biharmonic problem in the unit ball of $\mathbb{R}^{4}$. The weight is of logarithm type. The nonlinearity is critical in view of Adam's inequalities in the weighted Sobolev space $W^{2,2}_{0}(B,w)$. We prove the existence of non trivial solutions via the critical point theory. The main difficulty is the loss of compactness due to the critical exponential growth of the nonlinear term $f$. We give a new growth condition and we point out its importance for checking the Palais-Smale compactness condition.
We consider the fourth order Schrödinger operator $H=Δ^2+V$ and show that if there are no eigenvalues or resonances in the absolutely continuous spectrum of $H$ that the solution operator $e^{-itH}$ satisfies a large time integrable $|t|^{-\frac54}$ decay rate between weighted spaces. This bound improves what is possible for the free case in two directions; both better time decay and smaller spatial weights. In the case of a mild resonance at zero energy, we derive the operator-valued expansion $e^{-itH}P_{ac}(H)=t^{-\frac34} A_0+t^{-\frac54}A_1$ where $A_0:L^1\to L^\infty$ is an operator of r…
We investigate some focusing fourth-order coupled Schrodinger equations. Existence of ground state and global well-posedness are obtained. Moreover, the best constant of some Gagliardo-Nirenberg inequality is studied.
In this paper, we establish a weighted Adams' inequality in some appropriate weighted Sobolev space in $\mathbb{R}^4$. Then we give an improvement inequality by proving the concentration-compactness result. In the last part, we consider an application to elliptic equation involving new exponential growth at infinity in $\mathbb{R}^4$.
The goal of this work is to show that the generalized Korn inequality that replaces the symmetric part of the differential matrix in the classical Korn inequality by its trace-free part is valid over John domains and weighted Sobolev spaces. The weights considered are nonnegative powers of the distance to the boundary.
We present a weighted version of the Caffarelli-Kohn-Nirenberg inequality in the framework of variable exponents. The combination of this inequality with a variant of the fountain theorem, yields the existence of infinitely many solutions for a class of non-homogeneous problems with Dirichlet boundary condition.
A degenerate fourth-order parabolic equation modeling condensation phenomena related to Bose-Einstein particles is analyzed. The model is a Fokker-Planck-type approximation of the Boltzmann-Nordheim equation, only keeping the leading order term. It maintains some of the main features of the kinetic model, namely mass and energy conservation and condensation at zero energy. The existence of local-in-time weak solutions satisfying a certain entropy inequality is proven. The main result asserts that if a weighted $L^1$ norm of the initial data is sufficiently large and the initial data satisfies …
In this paper, we establish a weighted Trudinger-Moser type inequality with the full Sobolev norm constraint on the whole Euclidean space. Main tool is the singular Trudinger-Moser inequality on the whole space recently established by Adimurthi and Yang, and a transformation of functions. We also discuss the existence and non-existence of maximizers for the associated variational problem.
In this article, we prove a weighted Hardy inequality for $1<p<+\infty$ and dimension $d\ge1$. If $p>d$, then we can deduce from our weighted Hardy inequality a Poincaré inequality. The proof of the weighted Hardy inequality is based on the method of vector fields firstly introduced by Mitidieri \cite{MR1769903}. By the same method, we show for $1<p<+\infty$ and $d\ge1$ that weighted Caffarelli-Kohn-Nirenberg inequalities hold true. As an application of our weighted Hardy inequality, we prove a non-existence result for a p-Kolmogorov parabolic equation.
We first prove a weighted inequality of Moser-Trudinger type depending on a parameter, in the two-dimensional Euclidean space. The inequality holds for radial functions if the parameter is larger than -1. Without symmetry assumption, it holds if and only if the parameter is in the interval (-1,0]. The inequality gives us some insight on the symmetry breaking phenomenon for the extremal functions of the Hardy-Sobolev inequality, as established by Caffarelli-Kohn-Nirenberg, in two space dimensions. In fact, for suitable sets of parameters (asymptotically sharp) we prove symmetry or symmetry brea…
Let $Ω$ be a smooth bounded domain in $\mahbb R^N$ with $N\ge 3$ and let $Σ_k$ be a closed smooth submanifold of $δΩ$ of dimension $1\le k\le N-2$. In this paper we study the weighted Hardy inequality with weight function singular on $Σ_k$. In particular we provide sufficient and necessary conditions for existence of minimizers.
We establish the Trudinger-Moser inequality on weighted Sobolev spaces in the whole space, and for a class of quasilinear elliptic operators in radial form of the type $\displaystyle Lu:=-r^{-θ}(r^α\vert u'(r)\vert^βu'(r))'$, where $θ, β\geq 0$ and $α>0$, are constants satisfying some existence conditions. It worth emphasizing that these operators generalize the $p$- Laplacian and $k$-Hessian operators in the radial case. Our results involve fractional dimensions, a new weighted Pólya-Szeg{ö} principle, and a boundness value for the optimal constant in a Gagliardo-Nirenberg type inequality.
We show a weighted version of Korn inequality on bounded euclidean John domains, where the weights are nonnegative powers of the distance to the boundary. In this theorem, we also provide an estimate of the constant involved in the inequality which depends on the power that appears in the weight and a geometric condition that characterizes John domains. The proof uses a local-to-global argument based on a certain decomposition of functions. In addition, we prove the solvability in weighted Sobolev spaces of div(u)=f on the same class of domains. In this case, the weights are nonpositive powers…
In this paper, we establish the linear profile decomposition for the one dimensional fourth order Schrödinger equation $$ iu_t-μΔu+Δ^2u=0, t\in\mathbb{R}, x\in\mathbb{R}, u(0,x)=f(x)\in L^2, $$ where $μ\ge 0$. As an application, we establish a dichotomy result on the existence of extremals to the symmetric Schrödinger Strichartz inequality.
Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $N\geq 3$ and we let $Σ$ to be a closed submanifold of dimension $1 \leq k \leq N-2. $ In this paper we study existence and non-existence of minimizers of Hardy inequality with weight function singular on $Σ$ within the framework of Brezis-Marcus-Shafrir. In particular we provide necessary and sufficient conditions for existence of minimizers.
We continue to study the local well-posedness for higher order Benjamin-Ono type equations, especially fourth order equations. The proof is based on the energy methods with correction terms. Although one of correction terms can eliminate the highest order derivative loss in the energy inequality, it may yield a lower order derivative loss than the worst term. In order to cancel this derivative loss, we define correction terms inductively.
We study the fourth order Schrödinger type differential inequality $-Δ^2 u + λV(x)u \geq a(x)u^q$ with $a,V\in L^1_{loc}(\mathbf{R}^N)$, both nonnegative, and $λ>0$. We consider nonnegative solutions without making any assumptions about their asymptotic behavior, and choosing appropriate test functions, we find a priori estimates for some type of subharmonic solutions. Finally, applying these a priori estimates, we proof a Liouville type result.
Let $c, k_1,..., k_N $ be non-negative numbers, and define a measure $μ$ in the wedge $W:= \{x\in \mathbb{R} ^N :\, x_i >0, i=1,...,N\} $ by $dμ= e^{c|x|^2} x_1 ^{k_1}...x_N ^{k_N} \, dx $. It is shown that among all measurable subsets of $W$ with fixed $μ$ -measure, the intersection of $W$ with a ball centered at the origin renders the weighted perimeter relative to $W$ a minimum.
A degenerate fourth-order parabolic equation modeling condensation phenomena related to Bose-Einstein particles is analyzed. The model is a Fokker-Planck-type approximation of the Boltzmann-Nordheim equation, only keeping the leading order term. It maintains some of the main features of the kinetic model, namely mass and energy conservation and condensation at zero energy. The existence of a local-in-time nonnegative continuous weak solution is proven. If the solution is not global, it blows up with respect to the $L^\infty$ norm in finite time. The proof is based on approximation arguments, i…
In a note published in 1925, G. H. Hardy stated the inequality \begin{equation*} \sum_{n=1}^\infty \left(\frac{1}{n}\sum_{k=1}^n a_k \right)^p \leq \left(\frac{p}{p-1}\right)^p \sum_{n=1}^\infty a_n^p, \end{equation*} for any non-negative sequence $\{a_n\}_{n \geq 1}$, and $p>1$. This inequality is known in the literature as the classical discrete Hardy inequality. It has been widely studied and several applications and new versions have been shown. In this work, we use a characterization of a weighted version of this inequality to exhibit a sufficient condition for the existence of solutions …
In this paper, we show a weighted Hardy inequality in a limiting case for functions in weighted Sobolev spaces with respect to an invariant measure. We also prove that the constant in the left-hand side of the inequality is optimal. As applications, we establish the existence and nonexistence of positive exponentially bounded weak solutions to a parabolic problem involving the Ornstein-Uhlenbeck operator perturbed by a critical singular potential in two dimensional case, according to the size of the coefficient of the critical potential. These results can be considered as counterparts in the l…
We prove a sharp upper bound and a lower bound for the first nonzero eigenvalue of the Wentzell-Laplace operator on compact manifolds with boundary and an isoperimetric inequality for the same eigenvalue in the case where the manifold is a bounded domain in a Euclidean space. We study some fourth order Stekolv problems and obtain isoperimetric upper bound for the first eigenvalue of them. We also find all the eigenvalues and eigenfunctions for two kind of fourth order Stekolv problems on a Euclidean ball.
This paper is concerned with derivation of the global or local in time Strichartz estimates for radially symmetric solutions of the free wave equation from some Morawetz-type estimates via weighted Hardy-Littlewood-Sobolev (HLS) inequalities. In the same way we also derive the weighted end-point Strichartz estimates with gain of derivatives for radially symmetric solutions of the free Schroedinger equation. The proof of the weighted HLS inequality for radially symmetric functions involves an application of the weighted inequality due to Stein and Weiss and the Hardy-Littlewood maximal inequali…
In this paper we prove the fractional Gagliardo-Nirenberg inequality on homogeneous Lie groups. Also, we establish weighted fractional Caffarelli-Kohn-Nirenberg inequality and Lyapunov-type inequality for the Riesz potential on homogeneous Lie groups. The obtained Lyapunov inequality for the Riesz potential is new already in the classical setting of $\mathbb{R}^{N}$. As an application, we give two-sided estimate for the first eigenvalue of the Riesz potential. Also, we obtain Lyapunov inequality for the system of the fractional $p$-sub-Laplacian equations and give an application to estimate it…
A weighted relative isoperimetric inequality in convex cones is obtained via the Monge-Ampere equation. The method improves several inequalities in the literature, e.g. constants in a theorem of Cabre--Ros--Oton--Serra. Applications are given in the context of a generalization of the log-convex density conjecture due to Brakke and resolved by Chambers: in the case of $α-$homogeneous ($α>0$), concave densities, (mod translations) balls centered at the origin and intersected with the cone are proved to uniquely minimize the weighted perimeter with a weighted mass constraint. In particular, if th…
By using a suitable transform related to Sobolev inequality, we investigate the sharp constants and optimizers in radial space for the following weighted Caffarelli-Kohn-Nirenberg-type inequalities: \begin{equation*} \int_{\mathbb{R}^N}|x|^α|Δu|^2 dx \geq S^{rad}(N,α)\left(\int_{\mathbb{R}^N}|x|^{-α}|u|^{p^*_α} dx\right)^{\frac{2}{p^*_α}}, \quad u\in C^\infty_c(\mathbb{R}^N), \end{equation*} where $N\geq 3$, $4-N<α<2$, $p^*_α=\frac{2(N-α)}{N-4+α}$. Then we obtain the explicit form of the unique (up to scaling) radial positive solution $U_{λ,α}$ to the weighted fourth-order Hardy (for $α>0$) or…
A nonlinear fourth-order parabolic equation in one space dimension with periodic boundary conditions is studied. This equation arises in the context of fluctuations of a stationary nonequilibrium interface and in the modeling of quantum semiconductor devices. The existence of global-in-time non-negative weak solutions is shown. A criterion for the uniqueness of non-negative weak solutions is given. Finally, it is proved that the solution converges exponentially fast to its mean value in the ``entropy norm'' using a new optimal logarithmic Sobolev inequality for higher derivatives.
For a given Finsler-Minkowski norm $\mathcal{F}$ in $\mathbb{R}^N$ and a bounded smooth domain $Ω\subset\mathbb{R}^N$ $\big(N\geq 2\big)$, we establish the following weighted anisotropic Sobolev inequality $$ S\left(\int_Ω|u|^q f\,dx\right)^\frac{1}{q}\leq\left(\int_Ω\mathcal{F}(\nabla u)^p w\,dx\right)^\frac{1}{p},\quad\forall\,u\in W_0^{1,p}(Ω,w)\leqno{\mathcal{(P)}} $$ where $W_0^{1,p}(Ω,w)$ is the weighted Sobolev space under a class of $p$-admissible weights $w$, where $f$ is some nonnegative integrable function in $Ω$. We discuss the case $0<q<1$ and observe that $$ μ(Ω):=\inf_{u\in W_{0…
Given $(M, g)$ a smooth compact $(n+1)$-dimensional Riemannian manifold with boundary $\partial M$. Let $ρ$ be a defining function of $M$ and $σ\in(0,1)$. In this paper we study a weighted Sobolev-Poincaré type trace inequality corresponding to the embedding of $W^{1,2}(ρ^{1-2 σ}, M) \hookrightarrow L^{p}(\partial M)$, where $p=\frac{2 n}{n-2 σ}$. More precisely, under some assumptions on the manifold, we prove that there exists a constant $B>0$ such that, for all $u \in W^{1,2}(ρ^{1-2σ}, M)$, $$ \Big(\int_{\partial M}|u|^{p} \,\ud s_{g}\Big)^{2/p} \leq μ^{-1} \int_{M} ρ^{1-2 σ}|\nabla_{g} u|^…
In this paper we will establish different weighted Poincaré inequalities with variable exponents on Carnot-Carathéodory spaces or Carnot groups. We will use different techniques to obtain these inequalities. For vector fields satisfying Hörmander's condition in variable non-isotropic Sobolev spaces, we consider a weight in the variable Muckenhoupt class $% A_{p(\cdot ),p^{\ast }(\cdot )}$, where the exponent $p(\cdot )$ satisfies appropriate hypotheses, and in this case we obtain the first order weighted Poincaré inequalities with variable exponents. In the case of Carnot groups we also set up…
We prove a local two-weight Poincaré inequality for cubes using the sparse domination method that has been influential in harmonic analysis. The proof involves a localized version of the Fefferman--Stein inequality for the sharp maximal function. By establishing a local-to-global result in a bounded domain satisfying a Boman chain condition, we show a two-weight $p$-Poincaré inequality in such domains. As an application we show that certain nonnegative supersolutions of the $p$-Laplace equation and distance weights are $p$-admissible in a bounded domain, in the sense that they support versions…
Inspired by a recent pointwise differential inequality for positive bounded solutions of the fourth-order Hénon equation $Δ^2 u = |x|^a u^p$ in ${\mathbb R}^n$ with $a \geqslant 0$, $p > 1$, $n \geqslant 5$ due to Fazly, Wei, and Xu [ Anal. PDE., 8(2015) 1541--1563], first for some positive constants $α$ and $β$ we establish the following pointwise inequality \[ Δu \geqslant αu^{-\frac{q-1}2} + βu^{-1} |\nabla u|^2 \] in ${\mathbb R}^n$ with $n \geqslant 3$ for positive $C^4$-solutions of the fourth-order equation \[ Δ^2u=-u^{-q} \quad \text{ in } \mathbb R^n \] where $q > 1$. Next, we prove a…
We consider weighted $L^p$-Hardy inequalities involving the distance to the boundary of a domain in the $n$-dimensional Euclidean space with nonempty boundary. Using criticality theory, we give an alternative proof of the following result of F.~G.~Avkhadiev (2006) Theorem: Let $Ω\subsetneqq \mathbb{R}^n$, $n\geq 2$, be an arbitrary domain, $1<p<\infty$ and $α+ p>n$. Let $\mathrm{d}_Ω(x) =\mathrm{dist}(x,\partial Ω)$ denote the distance of a point $x\in Ω$ to $\partial Ω$. Then the following Hardy-type inequality holds $$ \int_{Ω}\frac{|\nabla φ|^p}{\mathrm{d}_Ω^α}\,\mathrm{d}x \geq \left( \fra…
We obtain heat kernel estimates for a class of fourth order non-uniformly elliptic operators in two dimensions. Contrary to existing results, the operators considered have symbols that are not strongly convex. This rises certain difficulties as it is known that, as opposed to the strongly convex case, there is no absolute exponential constant. Our estimates involve sharp constants and Finsler-type distances that are induced by the operator symbol. The main result is based on two general hypotheses, a weighted Sobolev inequalitry and an interpolation inequality, which are related to the singula…
In this paper, we consider the relation between $p > 1$ and critical dimension of the extremal solution of the semilinear equation $$\{\begin{array}{lllllll} βΔ^{2}u-τΔu=\fracλ{(1-u)^{p}} & in\ \ B, 0<u\leq 1 & in\ \ B, u=Δu=0 & on\ \ \partial B, \end{array} . $$ where $B$ is the unit ball in $R^{n}$, $λ>0$ is a parameter, $τ>0, β>0,p>1$ are fixed constants. By Hardy-Rellich inequality, we find that when $p$ is large enough, the critical dimension is 13.}
The well known duality between the Sobolev inequality and the Hardy-Littlewood-Sobolev inequality suggests that the Nash inequality could also have an interesting dual form, even though the Nash inequality relates three norms instead of two. We provide such a dual form here with sharp constants. This dual inequality relates the $L^2$ norm to the infimal convolution of the $L^\infty $ and $H^{-1}$ norms. The computation of this infimal convolution is a minimization problem, which we solve explicitly, thus providing a new proof of the sharp Nash inequality itself. This proof, via duality, also y…
