Ball In Rn Research Articles

Existence of solutions for some weighted mean field equations in dimension [formula omitted

This paper is devoted to studying the existence of solutions for the following weighted mean field equation (0.1)−div(ωβ(x)|∇u(x)|N−2∇u(x))=λθ|u|θ−1euθ∫Beuθdx,inB;u=0,on∂B,where B=B1(0) is the unit ball in RN, θ=NN−(N−1)β, β∈[0,1) and ωβ(x) is of logarithmic type function. By showing the logarithmic Trudinger–Moser inequalities in the weighted Sobolev spaces, we establish the existence of solutions for problem (0.1) when λ is in some certain range.

Poisson—Bergman Type Operators on Lipschitz and Mixed Norm Spaces in the Real Ball

Boundedness of some Poisson-Bergman type operators is stated over the unit ball in ℝn. Forelli-Rudin type theorems are proved and bounded harmonic projections are found on Lipschitz and mixed norm spaces.

Finite time blow-up for a nonlocal in time nonlinear heat equation in an exterior domain

We consider the nonlocal in time nonlinear heat equation ∂tu−Δu=1Γ(1−γ)∫0t(t−s)−γ|u(s)|pds+w(x),(t,x)∈(0,∞)×Dc with Dirichlet boundary conditions u(t,x)=f(t,x)≥0, (t,x)∈(0,∞)×∂D, where D is the unit open ball in RN, N≥2, Dc is its complement, p>1, 0<γ<1 and w⁄≡0. We show that if u(0,⋅)≥0 and ∫DcH(x)w(x)dx>0, where H is a certain harmonic function on Dc, then for all p>1, we have no global solutions.

Embeddings for [formula omitted]-weakly differentiable functions on domains

We prove that the inhomogeneous estimate of vector fields on balls in Rn(∫B|Dk−1u|n/(n−1)dx)(n−1)/n⩽c(∫B|Au|+|u|dx) for all u∈C∞(B¯,RN) holds if and only if the linear, constant coefficient differential operator A of order k has finite dimensional null-space (FDN). This generalizes the Gagliardo–Nirenberg–Sobolev inequality on domains and provides the local version of the analogous homogeneous embedding in full-space(∫Rn|Dk−1u|n/(n−1)dx)(n−1)/n⩽c∫Rn|Au|dx for all u∈Cc∞(Rn,RN), proved by Van Schaftingen precisely for elliptic and cancelling (EC) operators, building on fundamental L1-estimates from the works of Bourgain and Brezis. We prove that FDN strictly implies EC and discuss the contrast between homogeneous and inhomogeneous estimates on both algebraic and analytic level.

Finite time blow up of compressible Navier-Stokes equations on half space or outside a fixed ball

In this paper, we consider the initial-boundary value problem to the compressible Navier-Stokes equations for ideal gases without heat conduction in the half space or outside a fixed ball in RN, with N≥1. We prove that any classical solutions (ρ,u,θ), in the class C1([0,T];Hm(Ω)), m>[N2]+2, with bounded from below initial entropy and compactly supported initial density, which allows to touch the physical boundary, must blow-up in finite time, as long as the initial mass is positive. This paper extends the classical result by Xin (1998) [19], in which the Cauchy problem is considered, to the case that with physical boundary.

Iterated Function Systems with the Weak Average Contraction Conditions

This paper concerns the chaos game of random iterated function systems. We consider a random iterated function system IFS() generated by a finite family of Lipschitz maps on a compact ball of ℝn with the weak average contraction condition and show that it admits a quasi-attractor satisfying the deterministic chaos game. In particular, these properties are preserved under small perturbations of the iterated function system IFS() with respect to the Lipschitz topology.

The nodal set of solutions to some elliptic problems: Singular nonlinearities

This paper deals with solutions to the equation−Δu=λ+(u+)q−1−λ−(u−)q−1in B1 where λ+,λ−>0, q∈(0,1), B1=B1(0) is the unit ball in RN, N≥2, and u+:=max⁡{u,0}, u−:=max⁡{−u,0} are the positive and the negative part of u, respectively. We extend to this class of singular equations the results recently obtained in [25] for sublinear and discontinuous equations, 1≤q<2, namely: (a) the finiteness of the vanishing order at every point and the complete characterization of the order spectrum; (b) a weak non-degeneracy property; (c) regularity of the nodal set of any solution: the nodal set is a locally finite collection of regular codimension one manifolds up to a residual singular set having Hausdorff dimension at most N−2 (locally finite when N=2). As an intermediate step, we establish the regularity of a class of not necessarily minimal solutions.The proofs are based on a priori bounds, monotonicity formulæ for a 2-parameter family of Weiss-type functionals, blow-up arguments, and the classification of homogeneous solutions.

Linear Interpolation on a Euclidean Ball in Rⁿ

For \(x^{(0)}\in{\mathbb R}^n, R&gt;0\), by \(B=B(x^{(0)};R)\) we denote a Euclidean ball in \({\mathbb R}^n\) given by~the inequality \(\|x-x^{(0)}\|\leq R\), \(\|x\|:=\left(\sum_{i=1}^n x_i^2\right)^{1/2}\). Put \(B_n:=B(0,1)\). We mean by \(C(B)\) the space of~continuous functions \(f:B\to{\mathbb R}\) with the norm \(\|f\|_{C(B)}:=\max_{x\in B}|f(x)|\) and by \(\Pi_1\left({\mathbb R}^n\right)\) the set of polynomials in \(n\) variables of degree \(\leq 1\), i.e. linear functions on \({\mathbb R}^n\). Let \(x^{(1)}, \ldots, x^{(n+1)}\) be the~vertices of \(n\)-dimensional nondegenerate simplex \(S\subset B\). The interpolation projector \(P:C(B)\to \Pi_1({\mathbb R}^n)\) corresponding to \(S\) is defined by the equalities \(Pf\left(x^{(j)}\right)=%f_j:=f\left(x^{(j)}\right).\) Denote by \(\|P\|_B\) the norm of \(P\) as an operator from \(C(B)\) into \(C(B)\). Let us define \(\theta_n(B)\) as minimal value of \(\|P\|\) under the condition \(x^{(j)}\in B\). In the paper, we obtain the formula to compute \(\|P\|_B\) making use of \(x^{(0)}\), \(R\), and coefficients of basic Lagrange polynomials of \(S\). In more details we study the case when \(S\) is a regular simplex inscribed into \(B_n\). In this situation, we prove that \(\|P\|_{B_n}=\max\{\psi(a),\psi(a+1)\},\) where \(\psi(t)=\frac{2\sqrt{n}}{n+1}\bigl(t(n+1-t)\bigr)^{1/2}+\bigl|1-\frac{2t}{n+1}\bigr|\) \((0\leq t\leq n+1)\) and integer \(a\) has the form \(a=\bigl\lfloor\frac{n+1}{2}-\frac{\sqrt{n+1}}{2}\bigr\rfloor.\) For this projector, \(\sqrt{n}\leq\|P\|_{B_n}\leq\sqrt{n+1}\). The equality \(\|P\|_{B_n}=\sqrt{n+1}\) takes place if and only if \(\sqrt{n+1}\) is an integer number. We give the precise values of \(\theta_n(B_n)\) for \(1\leq n\leq 4\). To supplement theoretical results we present computational data. We also discuss some other questions concerning interpolation on a Euclidean ball.

Neumann eigenvalue problems on the exterior domains

For p∈(1,∞), we consider the following weighted Neumann eigenvalue problem on B1c, the exterior of the closed unit ball in RN: (0.1)−Δpϕ=λg|ϕ|p−2ϕinB1c,∂ϕ∂ν=0on∂B1,where Δp is the p-Laplace operator and g∈Lloc1(B1c) is an indefinite weight function. Depending on the values of p and the dimension N, we take g in certain Lorentz spaces or weighted Lebesgue spaces and show that (0.1) admits an unbounded sequence of positive eigenvalues that includes a unique principal eigenvalue. For this purpose, we establish the compact embeddings of W1,p(B1c) into Lp(B1c,|g|) for g in certain weighted Lebesgue spaces. For N>p, we also provide an alternate proof for the embedding of W1,p(B1c) into the Lorentz space Lp∗,p(B1c). Further, we show that the set of all eigenvalues of (0.1) is closed.

A sharp Trudinger-Moser type inequality involving Ln norm in the entire space [formula omitted

Let W1,n(Rn) be the standard Sobolev space and ‖⋅‖n be the Ln norm on Rn. We establish a sharp form of the following Trudinger-Moser inequality involving the Ln normsup‖u‖W1,n(Rn)=1∫RnΦ(αn|u|nn−1(1+α‖u‖nn)1n−1)dx<+∞ for any 0≤α<1, where Φ(t)=et−∑n−2j=0tjj!, αn=nωn−11n−1 and ωn−1 is the n−1 dimensional surface measure of the unit ball in Rn. We also show that the above supremum is infinity for all α≥1. Moreover, we prove the supremum is attained, namely, there exists a maximizer for the above supremum when α>0 is sufficiently small. The proof is based on the method of blow-up analysis of the nonlinear Euler-Lagrange equations of the Trudinger-Moser functionals.Our results sharpen the recent work [19] in which they show that the above inequality holds in a weaker form when Φ(t) is replaced by a strictly smaller Φ⁎(t)=et−∑n−1j=0tjj! (note that Φ(t)=Φ⁎(t)+tn−1(n−1)!).

Sharp estimates of semistable radial solutions of k-Hessian equations

AbstractWe consider semistable, radially symmetric and increasing solutions of Sk(D2u) = g(u) in the unit ball of ℝn, where Sk(D2u) is the k-Hessian operator of u and g ∈ C1 is a general positive nonlinearity. We establish sharp pointwise estimates for such solutions in a proper weighted Sobolev space, which are optimal and do not depend on the specific nonlinearity g. As an application of these results, we obtain pointwise estimates for the extremal solution and its derivatives (up to order three) of the equation Sk(D2u) = λg(u), posed in B1, with Dirichlet data $u\arrowvert _{B_1}=0$, where g is a continuous, positive, nonincreasing function such that lim t→−∞g(t)/|t|k = +∞.

Radial solutions of the Dirichlet problem for a class of quasilinear elliptic equations arising in optometry

This paper deals with the quasilinear elliptic problem − div∇u∕1+|∇u|2+a(x)u=b(x)∕1+|∇u|2inB,u=0on∂B,where B is an open ball in RN, with N≥2, and a,b∈C1(B¯) are given radially symmetric functions, with a(x)≥0 in B. This class of anisotropic prescribed mean curvature equations appears in the description of the geometry of the human cornea, as well as in the modeling theory of capillarity phenomena for compressible fluids. Unlike all previous works published on these subjects, existence and uniqueness of solutions of the above problem are here analyzed in the case where the coefficients a,b are not necessarily constant and no sign condition is assumed on b.

Positive radial solutions for multiparameter Dirichlet systems with mean curvature operator in Minkowski space and Lane–Emden type nonlinearities

We deal with Dirichlet systems involving the mean curvature operator in Minkowski spaceM(w)=div(∇w1−|∇w|2) in a ball in RN. Using the fixed point index and the lower and upper solutions method, we first obtain the existence of positive solutions for a class of differential systems with a singular φ-Laplacian, subjected to homogeneous mixed boundary conditions. The main application of the general results concerns Dirichlet systems with Lane–Emden type nonlinearities in a superlinear case, depending on two parameters λ1,λ2. So, we prove that for such a system there exists a continuous curve Γ which separates the first quadrant in two disjoint unbounded open sets O1 and O2 such that the system has zero, at least one or at least two radial positive solutions, according to (λ1,λ2)∈O1, (λ1,λ2)∈Γ or (λ1,λ2)∈O2.

On the structure of the nodal set and asymptotics of least energy sign-changing radial solutions of the fractional Brezis–Nirenberg problem

In this paper we study the asymptotic and qualitative properties of least energy radial sign-changing solutions of the fractional Brezis–Nirenberg problem ruled by the s-Laplacian, in a ball of Rn, when s∈(0,1) and n>6s. As usual, λ is the (positive) parameter in the linear part in u. We prove that for λ sufficiently small such solutions cannot vanish at the origin, we show that they change sign at most twice and their zeros coincide with the sign-changes. Moreover, when s is close to 1, such solutions change sign exactly once. Finally we prove that least energy nodal solutions which change sign exactly once have the limit profile of a “tower of bubbles”, as λ→0+, i.e. the positive and negative parts concentrate at the same point with different concentration speeds.

Entire solutions of a mean curvature flow connecting two periodic traveling waves

We consider the mean curvature flow V=H in a cylinder Ω≔B×R, where B is the unit ball in RN, V is the normal velocity of a moving surface Γt:xN+1=u(x,t)(x∈B) and H is the mean curvature of Γt. Assume that the surface is radially symmetric and it contacts the boundary of Ω with a prescribed angle θ=θ(t,u). In case θ tends to two (spatially or temporally) periodic functions as u→±∞, we show that the problem has a unique entire solution U, which propagates from −∞ to ∞ and connects two (spatially or temporally) periodic traveling waves at t=−∞ and at t=∞.

The nodal set of solutions to some elliptic problems: Sublinear equations, and unstable two-phase membrane problem

We are concerned with the nodal set of solutions to equations of the form−Δu=λ+(u+)q−1−λ−(u−)q−1in B1 where λ+,λ−>0, q∈[1,2), B1=B1(0) is the unit ball in RN, N≥2, and u+:=max⁡{u,0}, u−:=max⁡{−u,0} are the positive and the negative part of u, respectively. This class includes, the unstable two-phase membrane problem (q=1), as well as sublinear equations for 1<q<2.We prove the following main results: (a) the finiteness of the vanishing order at every point and the complete characterization of the order spectrum; (b) a weak non-degeneracy property; (c) regularity of the nodal set of any solution: the nodal set is a locally finite collection of regular codimension one manifolds up to a residual singular set having Hausdorff dimension at most N−2 (locally finite when N=2); (d) a partial stratification theorem.Ultimately, the main features of the nodal set are strictly related with those of the solutions to linear (or superlinear) equations, with two remarkable differences. First of all, the admissible vanishing orders can not exceed the critical value 2/(2−q). At threshold, we find a multiplicity of homogeneous solutions, yielding the non-validity of any estimate of the (N−1)-dimensional measure of the nodal set of a solution in terms of the vanishing order.As a byproduct, we also prove the strong unique continuation property for the unstable obstacle problem, corresponding to the case λ−=0.The proofs are based on monotonicity formulæ for a 2-parameter family of Weiss-type functionals, blow-up arguments, and the classification of homogeneous solutions.

On the strict monotonicity of the first eigenvalue of the 𝑝-Laplacian on annuli

Let B 1 B_1 be a ball in R N \mathbb {R}^N centred at the origin and let B 0 B_0 be a smaller ball compactly contained in B 1 B_1 . For p ∈ ( 1 , ∞ ) p\in (1, \infty ) , using the shape derivative method, we show that the first eigenvalue of the p p -Laplacian in annulus B 1 ∖ B 0 ¯ B_1\setminus \overline {B_0} strictly decreases as the inner ball moves towards the boundary of the outer ball. The analogous results for the limit cases as p → 1 p \to 1 and p → ∞ p \to \infty are also discussed. Using our main result, further we prove the nonradiality of the eigenfunctions associated with the points on the first nontrivial curve of the Fučik spectrum of the p p -Laplacian on bounded radial domains.

The Fredholm alternative for the [formula omitted]-Laplacian in exterior domains

We investigate the Fredholm alternative for the p-Laplacian in an exterior domain which is the complement of the closed unit ball in RN (N≥2). By employing techniques of Calculus of Variations we obtain the multiplicity of solutions. The striking difference between our case and the entire space case is also discussed.

Finite-time blow-up for quasilinear degenerate Keller–Segel systems of parabolic–parabolic type

This paper deals with the quasilinear degenerate Keller–Segel systems of parabolic–parabolic type in a ball of RN (N≥2). In the case of non-degenerate diffusion, Cieślak–Stinner [3,4] proved that if q>m+2N, where m denotes the intensity of diffusion and q denotes the nonlinearity, then there exist initial data such that the corresponding solution blows up in finite time. As to the case of degenerate diffusion, it is known that a solution blows up if q>m+2N (see Ishida–Yokota [13]); however, whether the blow-up time is finite or infinite has been unknown. This paper gives an answer to the unsolved problem. Indeed, the finite-time blow-up of energy solutions is established when q>m+2N.

Inclusion Properties of Orlicz Spaces and Weak Orlicz Spaces Generated by Concave Functions

In this paper, we introduce Orlicz spaces generated by concave functions and establish some inclusion properties between them. In addition, we also obtain similar result for weak Orlicz spaces. One of the keys to prove our result is by estimating characteristic function of open balls in ℝn.