Embedding Inequalities Research Articles

Nonlinear degenerate elliptic equation with Hardy potential and critical parameter

Some embedding inequalities in Hardy–Sobolev spaces with general weight functions were proved, and a positive answer to an open problem raised by Brezis–Vázquez was given. In the weighted Hardy–Sobolev spaces, the existence of nontrivial (many) solutions to the corresponding nonlinear degenerated elliptic equations with Hardy potential and critical parameter under conditions weaker than Ambrosetti–Rabinowitz condition, was obtained.

Optimal Sobolev embeddings on $\mathbb{R}^n$

We study Sobolev-type embeddings involving rearrangement-invariant norms. In particular, we focus on the question when such embeddings are optimal. We concentrate on the case when the functions involved are defined on $\mathbb R\sp n$. This subject has been studied before, but only on bounded domains. We first establish the equivalence of the Sobolev embedding to a new type of inequality involving two integral operators. Next, we show this inequality to be equivalent to the boundedness of a certain Hardy operator on a specific new type of cone of positive functions. This Hardy operator is then used to provide optimal domain and range rearrangement-invariant norm in the embedding inequality. Finally, the limiting case of the Sobolev embedding on Rn is studied in detail.

Lower Bounds for an Integral Involving Fractional Laplacians and the Generalized Navier-Stokes Equations in Besov Spaces

When estimating solutions of dissipative partial differential equations in L p -related spaces, we often need lower bounds for an integral involving the dissipative term. If the dissipative term is given by the usual Laplacian −Δ, lower bounds can be derived through integration by parts and embedding inequalities. However, when the Laplacian is replaced by the fractional Laplacian (−Δ) α , the approach of integration by parts no longer applies. In this paper, we obtain lower bounds for the integral involving (−Δ) α by combining pointwise inequalities for (−Δ) α with Bernstein's inequalities for fractional derivatives. As an application of these lower bounds, we establish the existence and uniqueness of solutions to the generalized Navier-Stokes equations in Besov spaces. The generalized Navier-Stokes equations are the equations resulting from replacing −Δ in the Navier-Stokes equations by (−Δ) α .

On the Domain of Analyticity for Solutions of Second Order Analytic Nonlinear Differential Equations

The radius of analyticity of periodic analytic functions can be characterized by the decay of their Fourier coefficients. This observation has led to the use of so-called Gevrey norms as a simple way of estimating the time evolution of the spatial radius of analyticity of solutions to parabolic as well as non-parabolic partial differential equations. In this paper we demonstrate, using a simple, explicitly solvable model equation, that estimates on the radius of analyticity obtained by the usual Gevrey class approach do not scale optimally across a family of solutions, nor do they scale optimally as a function of the physical parameters of the equation. We attribute the observed lack of sharpness to a specific embedding inequality, and give a modified definition of the Gevrey norms which is shown to finally yield a sharp estimate on the radius of analyticity.

Critical dimensions and higher order Sobolev inequalities with remainder terms

Pucci and Serrin [21] conjecture that certain space dimensions behave ''critically'' in a semilinear polyharmonic eigenvalue problem. Up to now only a considerably weakened version of this conjecture could be shown. We prove that exactly in these dimensions an embedding inequality for higher order Sobolev spaces on bounded domains with an optimal embedding constant may be improved by adding a ''linear'' remainder term, thereby giving further evidence to the conjecture of Pucci and Serrin from a functional analytic point of view. Thanks to Brezis-Lieb [5] this result is already known for the space \( H_0^1 \) in dimension n=3; we extend it to the spaces \( H_0^K \) (K>1) in the ''presumably'' critical dimensions. Crucial tools are positivity results and a decomposition method with respect to dual cones.

Embedding the weighted Sobolev space W{sup l}{sub p}({omega};v) in the space L{sub p}({omega};{omega})

Several conditions on the weight functions v and {omega} are obtained that guarantee the embedding inequality. Classes of weights {omega} and v in which these conditions are both necessary and sufficient are described.

Embedding the weighted Sobolev space in the space

Several conditions on the weight functions and are obtained that guarantee the embedding inequality Classes of weights and in which these conditions are both necessary and sufficient are described.

Sharp estimates and finite speed of propagation for a Neumann problem in domains narrowing at infinity

We investigate the connection between the geometry of an unbounded domain $\Omega$ and the existence and qualitative behaviour of solutions to a degenerate doubly non linear parabolic equation posed in $\Omega$. The domain $\Omega$ is assumed to be ``narrowing'' at infinity in a suitable sense, so that it has infinite volume. On the boundary of $\Omega$ we prescribe a homogeneous Neumann condition. Among other results, we prove sharp estimates for the finite speed of propagation of the support of positive solutions originating from initial data with bounded support. This is done by means of a new approach, which is flexible enough to be applied to the geometry at hand, and to cover the case of initial data measures. We also show that even if the initial datum has finite mass, the solution need not be globally bounded over $\Omega$ for a fixed positive time. We provide sharp estimates for such solutions. Our main tool is a new embedding inequality connected with the geometry of $\Omega$.

Inequalities related toH p smoothness of Sobolev type

By exploiting a class of maximal functions and Littlewood-Paley theory, a list of embedding inequalities onHp-Sobolev spaces andHp boundedness results for Riesz and Bessel potentials are obtained at one stroke.

Sharp Sobolev inequalities with interior norms

In this paper, we establish some new sharp Sobolev inequalities on any smooth bounded domain \(\Omega \subset{\Bbb R}^n\). Let \(S_1\) and S be the sharp constants corresponding to the Sobolev embedding and trace inequalities respectively. If \(n\ge 4\), there exist constants \(A(\Omega)\), \(A_1(\Omega)>0\) such that \(\forall u \in H^1(\Omega)\)\(\) and \(\) If \(n=3\), for any \(k_3 >3\), there exist constants \(A(\Omega, k_3), A_1(\Omega, k_3)>0 \) such that \(\forall u \in H^1(\Omega)\)\(\) and \(\)

Existence of solutions for a class of semilinear polyharmonic equations with critical exponential growth

The author considers the semilinear elliptic equation $$ -\Delta^{m}u=g(x,u), $$ subject to Dirichlet boundary conditions $u=Du=\cdots=D^{m-1}u=0$, on a bounded domain $\Omega\subset\mathbb{R}^{2m}$. The notion of nonlinearity of critical growth for this problem is introduced. It turns out that the critical growth rate is of exponential type and the problem is closely related to the Trudinger embedding and Moser type inequalities. The main result is the existence of non trivial weak solutions to the problem.

A nonlinear energy stability analysis of convection with temperature dependent viscosity

We establish a nonlinear energy stability theory for convection, when the viscosity depends on the temperature. The linearized stability problem is shown to be intimately connected with the nonlinear one, and the nonlinear analysis requires an intricate use of embedding inequalities. Numerical results are given for the nonlinear stability threshold and these represent rigorous practical bounds which should prove useful in experimental investigations of sub-critical instabilities in convection in temperature-dependent viscosity fluids.

Nonlinear stability for thermal convection in a micropolar fluid with temperature dependent viscosity

We establish a nonlinear energy stability theory for convection with the thermo-micropolar fluid of Eringen [1], allowing the viscosity to depend on the temperature. Due to the complicated nonlinearities which arise it is necessary to use a generalized energy analysis which requires an intricate use of embedding inequalities.