Ball In Rn Research Articles

Riesz and Kolmogorov inequality for harmonic quasiregular mappings

Let K⩾1 and p∈(1,2]. We obtain an asymptotically sharp constant c(K,p) as K→1 in the inequality‖ℑf‖p≤c(K,p)‖ℜf‖p, where f∈hp is a K-quasiregular harmonic mapping in the unit disk belonging to the Hardy space hp. This result holds under the conditions arg⁡(f(0))∈(−π2p,π2p) and f(D)∩(−∞,0)=∅. Our findings improve a recent result by Liu and Zhu [12]. Additionally, we extend this result to K-quasiregular harmonic mappings in the unit ball in Rn. Finally, we consider the Kolmogorov theorem for quasiregular harmonic mappings in the plane.

A bifurcation diagram of solutions to semilinear elliptic equations with general supercritical growth

We study the global bifurcation diagram of the positive solutions to the problem{Δu+λf(u)=0inB,u=0on∂B, where B is the unit ball in RN with N≥3. Under general supercritical growth conditions on f(u), we show that an unbounded bifurcation curve has no turning point, which indicates the existence of the singular extremal solution. In particular, our theory can be applied to the super-exponential cases of f(u), and we exhibit that a bifurcation curve for Δu+λf(u)=0 has the same qualitative property as a classical Gel'fand problem Δu+λeu=0 for N≥3 except N=10. Main technical tools are intrinsic transformations for semilinear elliptic equations and ODE techniques.

Infinitely many solutions for Hamiltonian system with critical growth

Abstract In this article, we consider the following elliptic system of Hamiltonian-type on a bounded domain: − Δ u = K 1 ( ∣ y ∣ ) ∣ v ∣ p − 1 v , in B 1 ( 0 ) , − Δ v = K 2 ( ∣ y ∣ ) ∣ u ∣ q − 1 u , in B 1 ( 0 ) , u = v = 0 on ∂ B 1 ( 0 ) , \left\{\begin{array}{ll}-\Delta u={K}_{1}\left(| y| ){| v| }^{p-1}v,\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{B}_{1}\left(0),\\ -\Delta v={K}_{2}\left(| y| ){| u| }^{q-1}u,\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{B}_{1}\left(0),\\ u=v=0\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial {B}_{1}\left(0),\end{array}\right. where K 1 ( r ) {K}_{1}\left(r) and K 2 ( r ) {K}_{2}\left(r) are positive bounded functions defined in [ 0 , 1 ] \left[0,1] , B 1 ( 0 ) {B}_{1}\left(0) is the unit ball in R N {{\mathbb{R}}}^{N} , and ( p , q ) \left(p,q) is a pair of positive numbers lying on the critical hyperbola 1 p + 1 + 1 q + 1 = N − 2 N . \frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}. Under some suitable further assumptions on the functions K 1 ( r ) {K}_{1}\left(r) and K 2 ( r ) {K}_{2}\left(r) , we prove the existence of infinitely many nonradial positive solutions whose energy can be made arbitrarily large. Our proof is based on the reduction method. The most ingredients of the article are using the Green representation and estimating the Green function and its regular part very carefully. For this purpose, some more extra ideas and techniques are needed. We believe that our method and techniques can be applied to other related problems.

Global properties of generic real–analytic nearly–integrable Hamiltonian systems

We introduce a new class Gsn of generic real analytic potentials on Tn and study global analytic properties of natural nearly–integrable Hamiltonians 12|y|2+εf(x), with potential f∈Gsn, on the phase space M=B×Tn with B a given ball in Rn. The phase space M can be covered by three sets: a ‘non–resonant’ set, which is filled up to an exponentially small set of measure e−cK (where K is the maximal size of resonances considered) by primary maximal KAM tori; a ‘simply resonant set’ of measure εKa and a third set of measure εKb which is ‘non perturbative’, in the sense that the H–dynamics on it can be described by a natural system which is not nearly–integrable. We then focus on the simply resonant set – the dynamics of which is particularly interesting (e.g., for Arnol'd diffusion, or the existence of secondary tori) – and show that on such a set the secular (averaged) 1 degree–of–freedom Hamiltonians (labeled by the resonance index k∈Zn) can be put into a universal form (which we call ‘Generic Standard Form’), whose main analytic properties are controlled by only one parameter, which is uniform in the resonance label k.

Existence results for the [formula omitted]-Hessian type system with the gradients via [formula omitted]-monotone matrices

In this work, we deal with the k-Hessian type system with the gradients SkσD2ui+α|∇ui|I=φi(|x|,−u1,−u2,…,−un), inΩ,ui=0, on∂Ω,i=1,2,…,n,where α≥0, n≥2, 1≤k≤N is a positive integer, I is the identity matrix and Ω stands for the open unit ball in RN(N≥2). Based on appropriate assumptions about φi(i=1,2,…,n), some results regarding existence of negative k-convex radial solution are established. More precisely, at least one solution and at least two solutions are obtained via the R+n-monotone matrices and the fixed point theory. Some basic inequality techniques such as Jensen inequality are applied, which allows us to overcome the difficulty that the k-Hessian type operator is related to the gradient terms. Finally, several examples are provided to show the validity of our main results.

Sign-changing solution for an overdetermined elliptic problem on unbounded domain

Abstract We prove the existence of two smooth families of unbounded domains in ℝ N + 1 {\mathbb{R}^{N+1}} with N ≥ 1 {N\geq 1} such that { - Δ ⁢ u = λ ⁢ u in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , ∂ ν ⁡ u = const on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle{}{-}\Delta u&\displaystyle=\lambda u&&% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\\ \displaystyle\partial_{\nu}u&\displaystyle=\mathrm{const}&&\displaystyle% \phantom{}\text{on }\partial\Omega,\end{aligned}\right. admits a sign-changing solution. The domains bifurcate from the straight cylinder B 1 × ℝ {B_{1}\times\mathbb{R}} , where B 1 {B_{1}} is the unit ball in ℝ N {\mathbb{R}^{N}} . These results can be regarded as counterexamples to the Berenstein Conjecture on unbounded domain. Unlike most previous papers in this direction, a very delicate issue here is that there may be two-dimensional kernel space at some bifurcation point. Thus a Crandall–Rabinowitz-type bifurcation theorem from high-dimensional kernel space is also established to achieve the goal.

Asymptotic non-degeneracy of the multipeak solution for the supercritical Hénon equation

We consider the following Hénon equation with supercritical exponent: −Δu=K(|y|)up+ɛ,u>0, in B1(0),u=0, on ∂B1(0),where B1(0) is the unit ball in RN(N≥4), p=2∗−1,2∗=2NN−2, K(r)∈C1[0,1]. We prove the asymptotic non-degeneracy of multipeak solution constructed in Liu and Peng (2016) via the local Pohozaev identities.

Eigenvalue problems for singular p-Monge-Ampère equations

Using the eigenvalue theory of completely continuous operators, we are concerned with the determining values of μ, in which there are nontrivial radial solutions to the singular p-Monge-Ampère problem{det(D(|Du|p−2Du))=μh(|x|)f(−u)inΩ,u=0on∂Ω, where μ>0 is a parameter, Ω is the open unit ball in Rn. In addition, as an application, we derive some criteria for the existence of nontrivial radial solutions to the power-type problem of p-Monge-Ampère equations. The dependence of nontrivial radial solution uμ on the parameter μ is also considered.

Topology of real multi-affine hypersurfaces and a homological stability property

Let R be a real closed field. We prove that the number of semi-algebraically connected components of a real hypersurface in Rn defined by a multi-affine polynomial of degree d is bounded by 2d−1. This bound is sharp and is independent of n (as opposed to the classical bound of d(2d−1)n−1 on the Betti numbers of hypersurfaces defined by arbitrary polynomials of degree d in Rn due to Petrovskiĭ and Oleĭnik, Thom and Milnor). Moreover, we show there exists c>1, such that given a sequence (Bn)n>0 where Bn is a closed ball in Rn of positive radius, there exist hypersurfaces (Vn⊂Rn)n>0 defined by symmetric multi-affine polynomials of degree 4, such that ∑i⩽5bi(Vn∩Bn)>cn, where bi(⋅) denotes the i-th Betti number with rational coefficients. Finally, as an application of the main result of the paper we verify a representational stability conjecture due to Basu and Riener on the cohomology modules of symmetric real algebraic sets for a new and much larger class of symmetric real algebraic sets than known before.

Weighted critical Hénon equations with p-Laplacian on the unit ball in ℝN

Weighted critical Hénon equations with p-Laplacian on the unit ball in ℝN

A sharp Poincaré inequality for functions in 𝐖^{1,∞}(Ω;ℝ)

For each natural number n n and any bounded, convex domain Ω ⊂ R n \Omega \subset \mathbb {R}^n we characterize the sharp constant C ( n , Ω ) C(n,\Omega ) in the Poincaré inequality ‖ f − f ¯ Ω ‖ L ∞ ( Ω ; R ) ≤ C ( n , Ω ) ‖ ∇ f ‖ L ∞ ( Ω ; R ) \| f - \bar {f}_{\Omega }\|_{L^{\infty }(\Omega ;\mathbb {R})} \leq C(n,\Omega ) \|\nabla f\|_{L^{\infty }(\Omega ;\mathbb {R})} . Here, f ¯ Ω \bar {f}_{\Omega } denotes the mean value of f f over Ω \Omega . In the case that Ω \Omega is a ball B r B_r of radius r r in R n \mathbb {R}^n , we calculate C ( n , B r ) = C ( n ) r C(n,B_r)=C(n)r explicitly in terms of n n and a ratio of the volumes of the unit balls in R 2 n − 1 \mathbb {R}^{2n-1} and R n \mathbb {R}^n . More generally, we prove that C ( n , B r ( Ω ) ) ≤ C ( n , Ω ) ≤ n n + 1 d i a m ( Ω ) C(n,B_{r(\Omega )}) \leq C(n,\Omega ) \leq \frac {n}{n+1}\mathrm {diam}(\Omega ) , where B r ( Ω ) B_{r(\Omega )} is a ball in R n \mathbb {R}^n with the same n − n- dimensional Lebesgue measure as Ω \Omega . Both bounds are sharp, and the lower bound can be interpreted as saying that, among convex domains of equal measure, balls have the best, i.e. smallest, Poincaré constant.

Singular semilinear elliptic problems on unbounded domains in [formula omitted

We prove the compactness of the solution operator for a class of singular semilinear elliptic problems on the exterior of a ball in Rn, n≥3. Compactness of solution operators for similar problems in Rn, n≥2 is established. Further, using these compactness results and employing Schauder-Tychonoff fixed point theorem, we prove the existence of a positive solution to classes of semipositone problems with asymptotically linear reaction terms.

On a weighted elliptic equation of N-Kirchhoff type with double exponential growth

Abstract In this work, we study the weighted Kirchhoff problem − g ∫ B σ ( x ) ∣ ∇ u ∣ N d x div ( σ ( x ) ∣ ∇ u ∣ N − 2 ∇ u ) = f ( x , u ) in B , u > 0 in B , u = 0 on ∂ B , \left\{\begin{array}{ll}-g\left(\mathop{\displaystyle \int }\limits_{B}\sigma \left(x)| \nabla u\hspace{-0.25em}{| }^{N}{\rm{d}}x\right){\rm{div}}\left(\sigma \left(x)| \nabla u\hspace{-0.25em}{| }^{N-2}\nabla u)=f\left(x,u)& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}B,\\ u\gt 0& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}B,\\ u=0& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial B,\end{array}\right. where B B is the unit ball of R N {{\mathbb{R}}}^{N} , σ ( x ) = log e ∣ x ∣ N − 1 \sigma \left(x)={\left(\log \left(\frac{e}{| x| }\right)\right)}^{N-1} , the singular logarithm weight in the Trudinger-Moser embedding, and g g is a continuous positive function on R + {{\mathbb{R}}}^{+} . The nonlinearity is critical or subcritical growth in view of Trudinger-Moser inequalities. We first obtain the existence of a solution in the subcritical exponential growth case with positive energy by using minimax techniques combined with the Trudinger-Moser inequality. In the critical case, the associated energy does not satisfy the condition of compactness. We provide a new condition for growth, and we stress its importance to check the compactness level.

A class of supercritical Sobolev type inequalities with logarithm and related elliptic equations

In this paper, we first deduce the following Sobolev inequality with logarithmic term:(0.1)sup⁡{∫B|u|2⁎|ln⁡(τ+|u|)||x|βdx:u∈H0,rad1(B),‖∇u‖L2(B)=1}<∞, where β>0, τ≥0 are constants, B is the unit ball in RN, N≥3, and 2⁎=2N/(N−2) is the critical Sobolev exponent. Then we show that the supremum in (0.1) is attained when 0<β<min⁡{N/2,N−2} and 1≤τ<∞. The inequality (0.1) can be used to prove the existence of positive solution for the following supercritical problem:(0.2){−Δu=u2⁎−1(ln⁡(τ+u))|x|β+g(|x|,u),u>0inB,u=0on∂B, where g(r,u)∈C([0,1)×R) is a subcritical perturbation. As a consequence, we can deduce the existence of positive solution for the supercritical problem with non-power nonlinearity:(0.3){−Δu=u2⁎−1(ln⁡(τ+u))|x|β,u>0inB,u=0on∂B. This is somewhat surprising, because the problem (0.3) has no nontrivial solution by Pohozaev's identity if the variable exponent |x|β is replaced by any non-negative constant.

Blow-up phenomena for a chemotaxis system with flux limitation

In this paper we consider nonnegative solutions of the following parabolic-elliptic cross-diffusion system{ut=Δu−∇(uf(|∇v|2)∇v),0=Δv−μ+u,∫Ωv=0,μ:=1|Ω|∫Ωudx,u(x,0)=u0(x), in Ω×(0,∞), with Ω a ball in RN, N≥3 under homogeneous Neumann boundary conditions and f(ξ)=(1+ξ)−α, 0<α<N−22(N−1), which describes gradient-dependent limitation of cross diffusion fluxes. Under conditions on f and initial data, we prove that a solution which blows up in finite time in L∞-norm, blows up also in Lp-norm for some p>1. Moreover, a lower bound of blow-up time is derived.

Existence and multiplicity of radial solutions for a k-Hessian system

In this paper, we consider the following nonlinear k-Hessian system{Sk(σ(D2z1))=λb(|x|)φ(−z1,−z2), inΩ,Sk(σ(D2z2))=μh(|x|)ψ(−z1,−z2), inΩ,z1=z2=0, on∂Ω, where λ,μ>0, Ω stands for the open unit ball in RN and Sk(σ(D2z)) is the k-Hessian operator of z. Under the appropriate assumptions on φ and ψ, the existence and multiplicity of the k-convex radial solutions are obtained. The method of proving theorems is the Guo-Krasnosel'skii fixed point theorem in a cone.

Game-theoretic p-Laplace operator involving the gradient

The game-theoretic p-Laplacian operator is a version of classical variational p-Laplacian which is in connection with stochastic games called Tug-of-War with noise. The existence of positive singular and Holder continuous solutions of the game-theoretic p-Laplace operator involving the gradient in a small C2 perturbation of the unit ball in Rn are proved. Finally, a more case problem is introduced.

Some sharp Schwarz-Pick type estimates and their applications of harmonic and pluriharmonic functions

The purpose of this paper is to study the Schwarz-Pick type inequalities for harmonic or pluriharmonic functions. By analogy with the generalized Khavinson conjecture, we first give some sharp estimates of the norm of harmonic functions from the Euclidean unit ball in Rn into the unit ball of the real Minkowski space. Next, we give several sharp Schwarz-Pick type inequalities for pluriharmonic functions from the Euclidean unit ball in Cn or from the unit polydisc in Cn into the unit ball of the Minkowski space. Furthermore, we establish some sharp coefficient type Schwarz-Pick inequalities for pluriharmonic functions defined in the Minkowski space. Finally, we use the obtained Schwarz-Pick type inequalities to discuss the Lipschitz continuity, the Schwarz-Pick type lemmas of arbitrary order and the Bohr phenomenon of harmonic or pluriharmonic functions.

An Expository Lecture of María Jesús Chasco on Some Applications of Fubini’s Theorem

The usefulness of Fubini’s theorem as a measurement instrument is clearly understood from its multiple applications in Analysis, Convex Geometry, Statistics or Number Theory. This article is an expository paper based on a master class given by the second author at the University of Vigo and is devoted to presenting some Applications of Fubini’s theorem. In the first part, we present Brunn–Minkowski’s and Isoperimetric inequalities. The second part is devoted to the estimations of volumes of sections of balls in Rn.

Stationary sets of the mean curvature flow with a forcing term

Abstract We consider the flat flow solution to the mean curvature equation with forcing in ℝ n {\mathbb{R}^{n}} . Our main result states that tangential balls in ℝ n {\mathbb{R}^{n}} under a flat flow with a bounded forcing term will experience fattening, which generalizes the result in [N. Fusco, V. Julin and M. Morini, Stationary sets and asymptotic behavior of the mean curvature flow with forcing in the plane, preprint 2020, https://arxiv.org/abs/2004.07734] from the planar case to higher dimensions. Then, as in the planar case, we characterize stationary sets in ℝ n {\mathbb{R}^{n}} for a constant forcing term as finite unions of equisize balls with mutually positive distance.