Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds

Let M be a complete noncompact Riemannian manifold. We consider gradient estimates for the positive solutions to the following nonlinear parabolic equation $ \frac{\partial u}{\partial t} = \Delta _{f}u +au\,{\rm log}\, u + bu$ on ${M \times [0, + \infty)}Let M be a complete noncompact Riemannian manifold. We consider gradient estimates for the positive solutions to the following nonlinear parabolic equation

\frac?u?t = Dfu +au log u + bu \frac{\partial u}{\partial t} = \Delta _{f}u +au\,{\rm log}\, u + bu      2. Gradient Estimates for the Porous Medium Equations on Riemannian Manifolds      Guangyue Huang  Zhijie Huang  Haizhong Li 《Journal of Geometric Analysis》2013,23(4):1851-1875 In this paper we study gradient estimates for the positive solutions of the porous medium equation: $$u_t=\Delta u^m$$ where m>1, which is a nonlinear version of the heat equation. We derive local gradient estimates of the Li–Yau type for positive solutions of porous medium equations on Riemannian manifolds with Ricci curvature bounded from below. As applications, several parabolic Harnack inequalities are obtained. In particular, our results improve the ones of Lu, Ni, Vázquez, and Villani (in J. Math. Pures Appl. 91:1–19, 2009). Moreover, our results recover the ones of Davies (in Cambridge Tracts Math vol. 92, 1989), Hamilton (in Comm. Anal. Geom. 1:113–125, 1993) and Li and Xu (in Adv. Math. 226:4456–4491, 2011).      3. Gradient estimates for the elliptic and parabolic Lichnerowicz equations on compact manifolds      Xianfa Song  Lin Zhao 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2010,61(4):655-662 Let (M, g) be a smooth compact Riemannian manifold of dimension n ≥ 3. Denote ${\Delta_g=-{\rm div}_g\nabla}$ the Laplace–Beltrami operator. We establish some local gradient estimates for the positive solutions of the Lichnerowicz equation $$\Delta_gu(x)+h(x)u(x)=A(x)u^p(x)+\frac{B(x)}{u^q(x)}$$ on (M, g). Here, p, q ≥ 0, A(x), B(x) and h(x) are smooth functions on (M, g). We also derive the Harnack differential inequality for the positive solutions of $$u_t(x,t)+\Delta_gu(x,t)+h(x)u(x,t)=A(x)u^p(x,t)+\frac{B(x)}{u^q(x,t)}$$ on (M, g) with initial data u(x, 0) > 0.      4. Gradient Estimates for Some Semi-Linear Hypoelliptic Equations      Bin Qian  Li Chen 《Acta Appl Math》2013,124(1):1-17 Given some smooth vector fields X 1,X 2,…,X m on a compact manifold M, if they satisfy Hörmander’s condition, we establish global gradient estimates for the positive smooth solutions to the semi-linear hypoelliptic equations $$Lu+au\log u+bu=\partial_tu, \quad \mbox{on} \ M\times[0,\infty) $$ and $$Lu+au\log u+bu=0, \quad \mbox{on} \ M, $$ where a,b are constants, and $L=\sum_{i}X_{i}^{2}-X_{0}$ . We partially generalize the results of Cao and Yau (Math. Z. 211:485–504, 1992).      5. Uniqueness of Fokker-Planck equations for spin lattice systems(Ⅱ): Non-compact case      LEMLE Ludovic Dan  WANG Ran  WU LiMing 《中国科学 数学(英文版)》2014,57(1):161-172 We study the existence and uniqueness of solutions for a class of infinite-dimensional Fokker-Planck equations on the spin lattice systems M Z d,where the spin space M is a non-compact Riemannian manifold.The method is based on the Stroock-Varadhan’s martingale approach,some compactness results of the general theory developed by Ethier-Kurtz,and some a priori gradient estimates.      6. Symmetry breaking bifurcation from solutions concentrating on the equator of \mathbb{S}^N      Yasuhito Miyamoto 《Journal d'Analyse Mathématique》2013,121(1):353-381 We are concerned with the elliptic problem $${\varepsilon ^2}{\Delta _{{S^n}}}u - u + {u^p} = 0{\text{ in }}{S^n},u > 0{\text{ in }}{S^n}$$ , where ${\Delta _{{S^n}}}$ is the Laplace-Beltrami operator on $\mathbb{S}^n : = \left\{ {x \in \mathbb{R}^{n + 1} ;\left\| x \right\| = 1} \right\}\left( {n \geqslant 3} \right)$ , and p ? 2. We construct a smooth branch C of solutions concentrating on the equator S n ∩ {x n+1 = 0}. Using the Crandall-Rabinowitz bifurcation theorem, we show that C has infinitely many bifurcation points from which continua of nonradial solutions emanate. In applying the bifurcation theorem, we verify the transversality condition directly.      7. A sharp gradient estimate for the weighted p-Laplacian      Lin-feng Wang  Yue-ping Zhu 《高校应用数学学报(英文版)》2012,27(4):462-474 Let M be an n-dimensional complete noncompact Riemannian manifold with sectional curvature bounded from below, d?? = e h (x) dV (x) the weighted measure and ????,p the weighted p-Laplacian. In this paper we consider the non-linear elliptic equation $$ \Delta _{\mu ,p} u = - \lambda _{\mu ,p} |u|^{p - 2} u $$ for p ?? (1, 2). We derive a sharp gradient estimate for positive smooth solutions of this equation. As applications, we get a Harnack inequality and a Liouville type theorem..      8. Ground state and multiple solutions for a critical exponent problem      Z. Chen  N. Shioji  W. Zou 《NoDEA : Nonlinear Differential Equations and Applications》2012,19(3):253-277 We study the following Brezis?CNirenberg type critical exponent equation which is related to the Yamabe problem: $$-\Delta u=\lambda u+ |u|^{2^{\ast}-2}u, \quad u\in H_0^1 (\Omega),$$ where ?? is a smooth bounded domain in ${{\mathbb R}^N(N\ge3)}$ and 2* is the critical Sobolev exponent. We show that, if N ?? 5, this problem has at least ${\lceil\frac{N+1}{2}\rceil}$ pairs of nontrivial solutions for each fixed ?? ?? ??1, where ??1 is the first eigenvalue of ??? with the Dirichlet boundary condition. For N ?? 3, we give energy estimates from below for ground state solutions.      9. Liouville theorems for entire local minimizers of energies defined on the class L log L and for entire solutions of the stationary Prandtl-Eyring fluid model      Martin Fuchs  Guo Zhang 《Calculus of Variations and Partial Differential Equations》2012,44(1-2):271-295 If ${u: \mathbb{R}^{n} \to \mathbb{R}^{M}}$ locally minimizes the energy with density ${|\nabla u|\ln (1 + |\nabla u|)}$ , then we show that the boundedness of the function u already implies its constancy. The same is true in case n = M = 2 for entire solutions of the equations modelling the stationary flow of a so-called Prandtl-Eyring fluid. Moreover, in the variational setting we will present various extensions of the above mentioned Liouville theorem for entire local minimizers valid in any dimensions n and M.      10. Existence and non-existence of positive solutions of the scalar field system in R n,I      Henghui Zou 《Calculus of Variations and Partial Differential Equations》1996,4(3):219-248 Letn > 3 andΩ be either the entire spaceR n or a Euclidean ball in R n . Consider the following boundary value problem (I) $$\{ _{\Delta v - u + u^q = 0,}^{\Delta u - v + v^p = 0,} u,v > 0, x \in \Omega $$ with homogeneous Dirichlet boundary data (replaced byu, v → 0 as ¦x¦ → ∞ when Ω=R n ), where p > 1 and q > 1. In this paper, we investigate the question of existence and non-existence of solutions of (I) and prove that (I) admits a solution if and only if $$\frac{1}{{p + 1}} + \frac{1}{{q + 1}} > \frac{{n - 2}}{n}$$ . The existence on a ball and onR n are established by a variational approach and an approximation argument respectively. The Pohozaev identity is used to show non-existence onR n .      11. Singularity and blow-up estimates via Liouville-type theorems for Hardy–Hénon parabolic equations      Quoc Hung Phan 《Journal of Evolution Equations》2013,13(2):411-442 We consider the Hardy–Hénon parabolic equation ${u_t-\Delta u =|x|^a |u|^{p-1}u}$ with p > 1 and ${a\in \mathbb{R}}$ . We establish the space-time singularity and decay estimates, and Liouville-type theorems for radial and nonradial solutions. As applications, we study universal and a priori bound of global solutions as well as the blow-up estimates for the corresponding initial-boundary value problem.      12. Positive solutions of quasilinear elliptic inequalities on noncompact Riemannian manifolds      A. G. Losev  Yu. S. Fedorenko 《Mathematical Notes》2007,81(5-6):778-787 In this paper, we consider the generalized solutions of the inequality $$ - div(A(x,u,\nabla u)\nabla u) \geqslant F(x,u,\nabla u)u^q , q > 1,$$ on noncompact Riemannian manifolds. We obtain sufficient conditions for the validity of Liouville’s theorem on the triviality of the positive solutions of the inequality under consideration. We also obtain sharp conditions for the existence of a positive solution of the inequality ? Δu ≥ u q, q > 1, on spherically symmetric noncompact Riemannian manifolds.      13. Decay of the L∞-norm of solutions of Navier-Stokes equations in unbounded domains      Michael Wiegner 《Acta Appl Math》1994,37(1-2):215-219 We show that for nn? 4 the L∞-norm of weak solutions of the Navier-Stokes equations on ?n with generalized energy inequality decays like $\parallel u(t, \cdot )\parallel _\infty = O(t^{ - ({{n + 1)} \mathord{\left/ {\vphantom {{n + 1)} 2}} \right. \kern-0em} 2}} ),if(1 + | \cdot |)|u(0, \cdot )| \in L_1 $ and $$\int_{\mathbb{R}^n } {u(0,x)} dx = 0$$ . The same holds for strong solutions in all dimensions, if additionally u(0, ·) ε Lp p >n.      14. Flowers on Riemannian manifolds      Regina Rotman 《Mathematische Zeitschrift》2011,269(1-2):543-554 In this paper, we will present two upper bounds for the length of a smallest “flower-shaped” geodesic net in terms of the volume and the diameter of a manifold. Minimal geodesic nets are critical points of the length functional on the space of graphs immersed into a Riemannian manifold. Let M n be a closed Riemannian manifold of dimension n. We prove that there exists a minimal geodesic net that consists of one vertex and at most 2n ? 1 geodesic loops based at that vertex of total length ≤ 2n!d, where d is the diameter of M n . We also show that there exists a minimal geodesic net that consists of one vertex and at most ${3^{(n+1)^2}}$ loops of total length ${\leq2 (n+1)!^2 3^{(n+1)^3}\,Fill\,Rad\,M^n \leq2(n+1)!^{\frac{5}{2}}3^{(n+1)^3}(n+1)n^n vol(M^n)^{\frac{1}{n}}}$ , where Fill Rad M n denotes the filling radius and vol(M n ) denotes the volume of M n .      15. Isometric Immersions into {\mathbb{S}^m \times \mathbb{R}} and {\mathbb{H}^m \times \mathbb{R}} with High Codimensions      Qun Chen  Qing Cui 《Results in Mathematics》2010,57(3-4):319-333 In this paper, we obtain sufficient and necessary conditions for a simply connected Riemannian manifold (M n , g) to be isometrically immersed into ${\mathbb{S}^m \times \mathbb{R}}$ and ${\mathbb{H}^m \times \mathbb{R}}$ .      16. A Nonlinear Loaded Parabolic Equation and a Related Inverse Problem      Kozhanov  A. I. 《Mathematical Notes》2004,76(5-6):784-795 The solvability of the nonlocal-in-time boundary-value problem for the nonlinear parabolic equation $$u_t - \Delta u + c(\bar u(x,T))u = f(x,t),$$ where $\bar u(x,t) = \alpha (t)u(x,t) + \int_0^t {\beta (\tau )u(x,\tau )d\tau } $ for given functions $\alpha (t)$ and $\beta (t)$ , is studied. Existence and uniqueness theorems for regular solutions are proved; it is shown that the results obtained can be used to study the solvability of coefficient inverse problems.      17. Interior regularity operators      R. E. White 《Analysis Mathematica》1981,7(3):217-233 ПустьР - линейный диф ференциальный опера тор с достаточно гладкими коэффициентами. По определению,P явля ется оператором внут ренней регулярности на ω ?R n т огда и только тогда, когда \(u \in B_{p,k_{ - N} }^{loc} (\Omega )\) и ω′?ω из условия \(Pu \in B_{p,k_s }^{loc} (\Omega ')\) вытекает, что \(u \in B_{p,k_s k}^{loc} (\Omega ')\) , где ?N+1≦s≦N. Соотве тствующий пример: $$Pu = - \Delta u + u c k(\xi ) = \xi _1^2 + \ldots + \xi _n^2 + 1.$$ Указанные операторы характеризуются в ра боте в терминах априорных н еравенств. До? казывается также сущ ествование локальны х фундаментальных реш ений для оператора, со пряженного кP, а также его гладкос ть вне диагонали. Эти результаты являются аналогами соответствующих рез ультатов для гипоэлл иптических операторов.      18. Measure data problems,lower-order terms and interpolation effects      Agnese Di Castro  Giampiero Palatucci 《Annali di Matematica Pura ed Applicata》2014,193(2):325-358 We deal with the solutions to nonlinear elliptic equations of the form $$-{\rm div}\, a(x, Du) + g(x, u)=f$$ , with f being just a summable function, under standard growth conditions on g and a. We prove general local decay estimates for level sets of the gradient of solutions in turn implying very general estimates in rearrangement and non-rearrangement function spaces, up to Lorentz–Morrey spaces. The results obtained are in clear accordance with the classical Gagliardo–Nirenberg interpolation theory.      19. A priori estimates for a class of degenerate elliptic equations      L. H. de Miranda  M. Montenegro 《NoDEA : Nonlinear Differential Equations and Applications》2013,20(5):1683-1699 In this paper we investigate the regularity of solutions for the following degenerate partial differential equation $$\left \{\begin{array}{ll} -\Delta_p u + u = f \qquad {\rm in} \,\Omega,\\ \frac{\partial u}{\partial \nu} = 0 \qquad \qquad \,\,\,\,\,\,\,\,\,\, {\rm on} \,\partial \Omega, \end{array}\right.$$ when ${f \in L^q(\Omega), p > 2}$ and q ≥ 2. If u is a weak solution in ${W^{1, p}(\Omega)}$ , we obtain estimates for u in the Nikolskii space ${\mathcal{N}^{1+2/r,r}(\Omega)}$ , where r = q(p ? 2) + 2, in terms of the L q norm of f. In particular, due to imbedding theorems of Nikolskii spaces into Sobolev spaces, we conclude that ${\|u\|^r_{W^{1 + 2/r - \epsilon, r}(\Omega)} \leq C(\|f\|_{L^q(\Omega)}^q + \| f\|^{r}_{L^q(\Omega)} + \|f\|^{2r/p}_{L^q(\Omega)})}$ for every ${\epsilon > 0}$ sufficiently small. Moreover, we prove that the resolvent operator is continuous and compact in ${W^{1,r}(\Omega)}$ .      20. Blow-up solutions concentrated along minimal submanifolds for some supercritical elliptic problems on Riemannian manifolds      Marco Ghimenti  Anna Maria Micheletti  Angela Pistoia 《Journal of Fixed Point Theory and Applications》2013,14(2):503-525 Let (M, g) and \({(K, \kappa)}\) be two Riemannian manifolds of dimensions m and k, respectively. Let \({\omega \in C^{2} (N), \omega > 0}\) . The warped product \({M \times_\omega K}\) is the (m +  k)-dimensional product manifold \({M \times K}\) furnished with metric \({g + \omega^{2} \kappa}\) . We prove that the supercritical problem $$- \Delta_{g + \omega^{2} \kappa} u + hu = u^{\frac{m+2}{m-2} \pm \varepsilon} ,\quad u > 0,\quad {\rm in}\,\, (M \times_{\omega} K, g + \omega^{2} \kappa)$$ has a solution concentrated along a k-dimensional minimal submanifold \({\Gamma}\) of \({M \times_{\omega } N}\) as the real parameter \({\varepsilon}\) goes to zero, provided the function h and the sectional curvatures along \({\Gamma}\) satisfy a suitable condition.

Gradient Estimates for the Porous Medium Equations on Riemannian Manifolds

In this paper we study gradient estimates for the positive solutions of the porous medium equation: $$u_t=\Delta u^m$$ where m>1, which is a nonlinear version of the heat equation. We derive local gradient estimates of the Li–Yau type for positive solutions of porous medium equations on Riemannian manifolds with Ricci curvature bounded from below. As applications, several parabolic Harnack inequalities are obtained. In particular, our results improve the ones of Lu, Ni, Vázquez, and Villani (in J. Math. Pures Appl. 91:1–19, 2009). Moreover, our results recover the ones of Davies (in Cambridge Tracts Math vol. 92, 1989), Hamilton (in Comm. Anal. Geom. 1:113–125, 1993) and Li and Xu (in Adv. Math. 226:4456–4491, 2011).     

Gradient estimates for the elliptic and parabolic Lichnerowicz equations on compact manifolds

Let (M, g) be a smooth compact Riemannian manifold of dimension n ≥ 3. Denote ${\Delta_g=-{\rm div}_g\nabla}$ the Laplace–Beltrami operator. We establish some local gradient estimates for the positive solutions of the Lichnerowicz equation $$\Delta_gu(x)+h(x)u(x)=A(x)u^p(x)+\frac{B(x)}{u^q(x)}$$ on (M, g). Here, p, q ≥ 0, A(x), B(x) and h(x) are smooth functions on (M, g). We also derive the Harnack differential inequality for the positive solutions of $$u_t(x,t)+\Delta_gu(x,t)+h(x)u(x,t)=A(x)u^p(x,t)+\frac{B(x)}{u^q(x,t)}$$ on (M, g) with initial data u(x, 0) > 0.     

Gradient Estimates for Some Semi-Linear Hypoelliptic Equations

Given some smooth vector fields X ₁,X ₂,…,X _m on a compact manifold M, if they satisfy Hörmander’s condition, we establish global gradient estimates for the positive smooth solutions to the semi-linear hypoelliptic equations $$Lu+au\log u+bu=\partial_tu, \quad \mbox{on} \ M\times[0,\infty) $$ and $$Lu+au\log u+bu=0, \quad \mbox{on} \ M, $$ where a,b are constants, and $L=\sum_{i}X_{i}^{2}-X_{0}$ . We partially generalize the results of Cao and Yau (Math. Z. 211:485–504, 1992).     

Uniqueness of Fokker-Planck equations for spin lattice systems(Ⅱ): Non-compact case

We study the existence and uniqueness of solutions for a class of infinite-dimensional Fokker-Planck equations on the spin lattice systems M Z d,where the spin space M is a non-compact Riemannian manifold.The method is based on the Stroock-Varadhan’s martingale approach,some compactness results of the general theory developed by Ethier-Kurtz,and some a priori gradient estimates.     

Symmetry breaking bifurcation from solutions concentrating on the equator of \mathbb{S}^N

We are concerned with the elliptic problem $${\varepsilon ^2}{\Delta _{{S^n}}}u - u + {u^p} = 0{\text{ in }}{S^n},u > 0{\text{ in }}{S^n}$$ , where ${\Delta _{{S^n}}}$ is the Laplace-Beltrami operator on $\mathbb{S}^n : = \left\{ {x \in \mathbb{R}^{n + 1} ;\left\| x \right\| = 1} \right\}\left( {n \geqslant 3} \right)$ , and p ? 2. We construct a smooth branch C of solutions concentrating on the equator S ⁿ ∩ {x _{n+1 = 0}}. Using the Crandall-Rabinowitz bifurcation theorem, we show that C has infinitely many bifurcation points from which continua of nonradial solutions emanate. In applying the bifurcation theorem, we verify the transversality condition directly.     

A sharp gradient estimate for the weighted p-Laplacian

Let M be an n-dimensional complete noncompact Riemannian manifold with sectional curvature bounded from below, d?? = e ^h (x) dV (x) the weighted measure and ??_??,p the weighted p-Laplacian. In this paper we consider the non-linear elliptic equation $$ \Delta _{\mu ,p} u = - \lambda _{\mu ,p} |u|^{p - 2} u $$ for p ?? (1, 2). We derive a sharp gradient estimate for positive smooth solutions of this equation. As applications, we get a Harnack inequality and a Liouville type theorem..     

Ground state and multiple solutions for a critical exponent problem

We study the following Brezis?CNirenberg type critical exponent equation which is related to the Yamabe problem: $$-\Delta u=\lambda u+ |u|^{2^{\ast}-2}u, \quad u\in H_0^1 (\Omega),$$ where ?? is a smooth bounded domain in ${{\mathbb R}^N(N\ge3)}$ and 2^* is the critical Sobolev exponent. We show that, if N ?? 5, this problem has at least ${\lceil\frac{N+1}{2}\rceil}$ pairs of nontrivial solutions for each fixed ?? ?? ??₁, where ??₁ is the first eigenvalue of ??? with the Dirichlet boundary condition. For N ?? 3, we give energy estimates from below for ground state solutions.     

Liouville theorems for entire local minimizers of energies defined on the class L log L and for entire solutions of the stationary Prandtl-Eyring fluid model

If ${u: \mathbb{R}^{n} \to \mathbb{R}^{M}}$ locally minimizes the energy with density ${|\nabla u|\ln (1 + |\nabla u|)}$ , then we show that the boundedness of the function u already implies its constancy. The same is true in case n = M = 2 for entire solutions of the equations modelling the stationary flow of a so-called Prandtl-Eyring fluid. Moreover, in the variational setting we will present various extensions of the above mentioned Liouville theorem for entire local minimizers valid in any dimensions n and M.     

Existence and non-existence of positive solutions of the scalar field system in R n,I

Letn > 3 andΩ be either the entire spaceR ⁿ or a Euclidean ball in R ⁿ . Consider the following boundary value problem (I) $$\{ _{\Delta v - u + u^q = 0,}^{\Delta u - v + v^p = 0,} u,v > 0, x \in \Omega $$ with homogeneous Dirichlet boundary data (replaced byu, v → 0 as ¦x¦ → ∞ when Ω=R ⁿ ), where p > 1 and q > 1. In this paper, we investigate the question of existence and non-existence of solutions of (I) and prove that (I) admits a solution if and only if $$\frac{1}{{p + 1}} + \frac{1}{{q + 1}} > \frac{{n - 2}}{n}$$ . The existence on a ball and onR ⁿ are established by a variational approach and an approximation argument respectively. The Pohozaev identity is used to show non-existence onR ⁿ .     

Singularity and blow-up estimates via Liouville-type theorems for Hardy–Hénon parabolic equations

We consider the Hardy–Hénon parabolic equation ${u_t-\Delta u =|x|^a |u|^{p-1}u}$ with p > 1 and ${a\in \mathbb{R}}$ . We establish the space-time singularity and decay estimates, and Liouville-type theorems for radial and nonradial solutions. As applications, we study universal and a priori bound of global solutions as well as the blow-up estimates for the corresponding initial-boundary value problem.     

Positive solutions of quasilinear elliptic inequalities on noncompact Riemannian manifolds

In this paper, we consider the generalized solutions of the inequality $$ - div(A(x,u,\nabla u)\nabla u) \geqslant F(x,u,\nabla u)u^q , q > 1,$$ on noncompact Riemannian manifolds. We obtain sufficient conditions for the validity of Liouville’s theorem on the triviality of the positive solutions of the inequality under consideration. We also obtain sharp conditions for the existence of a positive solution of the inequality ? Δu ≥ u ^q, q > 1, on spherically symmetric noncompact Riemannian manifolds.     

Decay of the L∞-norm of solutions of Navier-Stokes equations in unbounded domains

We show that for nn? 4 the L_∞-norm of weak solutions of the Navier-Stokes equations on ?ⁿ with generalized energy inequality decays like $\parallel u(t, \cdot )\parallel _\infty = O(t^{ - ({{n + 1)} \mathord{\left/ {\vphantom {{n + 1)} 2}} \right. \kern-0em} 2}} ),if(1 + | \cdot |)|u(0, \cdot )| \in L_1 $ and $$\int_{\mathbb{R}^n } {u(0,x)} dx = 0$$ . The same holds for strong solutions in all dimensions, if additionally u(0, ·) ε L_p p >n.     

Flowers on Riemannian manifolds

In this paper, we will present two upper bounds for the length of a smallest “flower-shaped” geodesic net in terms of the volume and the diameter of a manifold. Minimal geodesic nets are critical points of the length functional on the space of graphs immersed into a Riemannian manifold. Let M ⁿ be a closed Riemannian manifold of dimension n. We prove that there exists a minimal geodesic net that consists of one vertex and at most 2n ? 1 geodesic loops based at that vertex of total length ≤ 2n!d, where d is the diameter of M ⁿ . We also show that there exists a minimal geodesic net that consists of one vertex and at most ${3^{(n+1)^2}}$ loops of total length ${\leq2 (n+1)!^2 3^{(n+1)^3}\,Fill\,Rad\,M^n \leq2(n+1)!^{\frac{5}{2}}3^{(n+1)^3}(n+1)n^n vol(M^n)^{\frac{1}{n}}}$ , where Fill Rad M ⁿ denotes the filling radius and vol(M ⁿ ) denotes the volume of M ⁿ .     

Isometric Immersions into {\mathbb{S}^m \times \mathbb{R}} and {\mathbb{H}^m \times \mathbb{R}} with High Codimensions

In this paper, we obtain sufficient and necessary conditions for a simply connected Riemannian manifold (M ⁿ , g) to be isometrically immersed into ${\mathbb{S}^m \times \mathbb{R}}$ and ${\mathbb{H}^m \times \mathbb{R}}$ .     

A Nonlinear Loaded Parabolic Equation and a Related Inverse Problem

The solvability of the nonlocal-in-time boundary-value problem for the nonlinear parabolic equation $$u_t - \Delta u + c(\bar u(x,T))u = f(x,t),$$ where $\bar u(x,t) = \alpha (t)u(x,t) + \int_0^t {\beta (\tau )u(x,\tau )d\tau } $ for given functions $\alpha (t)$ and $\beta (t)$ , is studied. Existence and uniqueness theorems for regular solutions are proved; it is shown that the results obtained can be used to study the solvability of coefficient inverse problems.     

Interior regularity operators

ПустьР - линейный диф ференциальный опера тор с достаточно гладкими коэффициентами. По определению,P явля ется оператором внут ренней регулярности на ω ?R ⁿ т огда и только тогда, когда \(u \in B_{p,k_{ - N} }^{loc} (\Omega )\) и ω′?ω из условия \(Pu \in B_{p,k_s }^{loc} (\Omega ')\) вытекает, что \(u \in B_{p,k_s k}^{loc} (\Omega ')\) , где ?N+1≦s≦N. Соотве тствующий пример: $$Pu = - \Delta u + u c k(\xi ) = \xi _1^2 + \ldots + \xi _n^2 + 1.$$ Указанные операторы характеризуются в ра боте в терминах априорных н еравенств. До^? казывается также сущ ествование локальны х фундаментальных реш ений для оператора, со пряженного кP, а также его гладкос ть вне диагонали. Эти результаты являются аналогами соответствующих рез ультатов для гипоэлл иптических операторов.     

Measure data problems,lower-order terms and interpolation effects

We deal with the solutions to nonlinear elliptic equations of the form $$-{\rm div}\, a(x, Du) + g(x, u)=f$$ , with f being just a summable function, under standard growth conditions on g and a. We prove general local decay estimates for level sets of the gradient of solutions in turn implying very general estimates in rearrangement and non-rearrangement function spaces, up to Lorentz–Morrey spaces. The results obtained are in clear accordance with the classical Gagliardo–Nirenberg interpolation theory.     

A priori estimates for a class of degenerate elliptic equations

In this paper we investigate the regularity of solutions for the following degenerate partial differential equation $$\left \{\begin{array}{ll} -\Delta_p u + u = f \qquad {\rm in} \,\Omega,\\ \frac{\partial u}{\partial \nu} = 0 \qquad \qquad \,\,\,\,\,\,\,\,\,\, {\rm on} \,\partial \Omega, \end{array}\right.$$ when ${f \in L^q(\Omega), p > 2}$ and q ≥ 2. If u is a weak solution in ${W^{1, p}(\Omega)}$ , we obtain estimates for u in the Nikolskii space ${\mathcal{N}^{1+2/r,r}(\Omega)}$ , where r = q(p ? 2) + 2, in terms of the L ^q norm of f. In particular, due to imbedding theorems of Nikolskii spaces into Sobolev spaces, we conclude that ${\|u\|^r_{W^{1 + 2/r - \epsilon, r}(\Omega)} \leq C(\|f\|_{L^q(\Omega)}^q + \| f\|^{r}_{L^q(\Omega)} + \|f\|^{2r/p}_{L^q(\Omega)})}$ for every ${\epsilon > 0}$ sufficiently small. Moreover, we prove that the resolvent operator is continuous and compact in ${W^{1,r}(\Omega)}$ .     

Blow-up solutions concentrated along minimal submanifolds for some supercritical elliptic problems on Riemannian manifolds

Let (M, g) and \({(K, \kappa)}\) be two Riemannian manifolds of dimensions m and k, respectively. Let \({\omega \in C^{2} (N), \omega > 0}\) . The warped product \({M \times_\omega K}\) is the (m +  k)-dimensional product manifold \({M \times K}\) furnished with metric \({g + \omega^{2} \kappa}\) . We prove that the supercritical problem $$- \Delta_{g + \omega^{2} \kappa} u + hu = u^{\frac{m+2}{m-2} \pm \varepsilon} ,\quad u > 0,\quad {\rm in}\,\, (M \times_{\omega} K, g + \omega^{2} \kappa)$$ has a solution concentrated along a k-dimensional minimal submanifold \({\Gamma}\) of \({M \times_{\omega } N}\) as the real parameter \({\varepsilon}\) goes to zero, provided the function h and the sectional curvatures along \({\Gamma}\) satisfy a suitable condition.