Uniqueness of self-shrinkers to the degree-one curvature flow with a tangent cone at infinity

Given a smooth, symmetric and homogeneous of degree one function \(f\left( \lambda _{1},\ldots ,\lambda _{n}\right) \) satisfying \(\partial _{i}f>0\quad \forall \,i=1,\ldots , n\), and a properly embedded smooth cone \({\mathcal {C}}\) in \({\mathbb {R}}^{n+1}\), we show that under suitable conditions on f, there is at most one f self-shrinker (i.e. a hypersurface \(\Sigma \) in \({\mathbb {R}}^{n+1}\) satisfying \(f\left( \kappa _{1},\ldots ,\kappa _{n}\right) +\frac{1}{2}X\cdot N=0\), where \(\kappa _{1},\ldots ,\kappa _{n}\) are principal curvatures of \(\Sigma \)) that is asymptotic to the given cone \({\mathcal {C}}\) at infinity.     

Three solutions for a perturbed Navier problem

In this paper we prove the existence of at least three distinct solutions to the following perturbed Navier problem:

$$\left\{\begin{array}{ll}\Delta (|{\Delta u}|^{p-2}\Delta u) = f(x,u) + \lambda g(x,u) \quad{\rm in}\,\,\,\Omega \\ u=\Delta u = 0 \qquad\qquad\qquad\qquad\qquad\quad{\rm on}\,\,\, \partial \Omega,\end{array}\right.$$

where \({{\Omega \subset \mathbb {R}^N}}\) is an open bounded set with smooth boundary \({\partial \Omega}\) and \({\lambda \in \mathbb {R}}\) . Under very mild conditions on g and some assumptions on the behaviour of the potential of f at 0 and +∞, our result assures the existence of at least three distinct solutions to the above problem for λ small enough. Moreover such solutions belong to a ball of the space \({W^{2,p}(\Omega)\cap W_0^{1,p}(\Omega)}\) centered in the origin and with radius not dependent on λ.

     

Generalized Joseph–Lundgren exponent and intersection properties for supercritical quasilinear elliptic equations

We study the solution \({u(r,\rho)}\) of the quasilinear elliptic problem

$$\begin{cases}r^{-(\gamma-1)}(r^{\alpha}|u'|^{\beta-1}u')'+|u|^{p-1}u=0, & 0 < r < \infty, \\u(0)=\rho > 0,\ u'(0)=0.\end{cases}$$

The usual Laplace, \({m}\)-Laplace, and \({k}\)-Hessian operators are included in the differential operator \({r^{-(\gamma-1)}(r^{\alpha}|u'|^{\beta-1}u')'}\). Under certain conditions on \({\alpha}\), \({\beta}\), \({\gamma}\), and \({p}\), the equation has a singular positive solution \({u^*(r)}\) and the solution \({u(r,\rho)}\) is positive for \({r\ge 0}\). We study the intersection numbers between \({u(r,\rho)}\) and \({u^*(r)}\) and between \({u(r,\rho_0)}\) and \({u(r,\rho_1)}\). A generalized Joseph–Lundgren exponent \({p^*_{JL}}\) plays a crucial role. The main technique is a phase plane analysis. In particular, we use two changes of variables which transform the equation into two autonomous systems.

     

Some remarks on energy inequalities for harmonic maps with potential

We call the \({\delta}\)-vector of an integral convex polytope of dimension d flat if the \({\delta}\)-vector is of the form \({(1,0,\ldots,0,a,\ldots,a,0,\ldots,0)}\), where \({a \geq 1}\). In this paper, we give the complete characterization of possible flat \({\delta}\)-vectors. Moreover, for an integral convex polytope \({\mathcal{P}\subset \mathbb{R}^N}\) of dimension d, we let \({i(\mathcal{P},n)=|n\mathcal{P}\cap \mathbb{Z}^N|}\) and \({i^*(\mathcal{P},n)=|n(\mathcal{P} {\setminus}\partial \mathcal{P})\cap \mathbb{Z}^N|}\). By this characterization, we show that for any \({d \geq 1}\) and for any \({k,\ell \geq 0}\) with \({k+\ell \leq d-1}\), there exist integral convex polytopes \({\mathcal{P}}\) and \({\mathcal{Q}}\) of dimension d such that (i) For \({t=1,\ldots,k}\), we have \({i(\mathcal{P},t)=i(\mathcal{Q},t),}\) (ii) For \({t=1,\ldots,\ell}\), we have \({i^*(\mathcal{P},t)=i^*(\mathcal{Q},t)}\), and (iii) \({i(\mathcal{P},k+1) \neq i(\mathcal{Q},k+1)}\) and \({i^*(\mathcal{P},\ell+1)\neq i^*(\mathcal{Q},\ell+1)}\).     

The Proper Dissipative Extensions of a Dual Pair

Let A and \({(-\widetilde{A})}\) be dissipative operators on a Hilbert space \({\mathcal{H}}\) and let \({(A,\widetilde{A})}\) form a dual pair, i.e. \({A \subset \widetilde{A}^*}\), resp. \({\widetilde{A} \subset A^*}\). We present a method of determining the proper dissipative extensions \({\widehat{A}}\) of this dual pair, i.e. \({A\subset \widehat{A} \subset\widetilde{A}^*}\) provided that \({\mathcal{D}(A)\cap\mathcal{D}(\widetilde{A})}\) is dense in \({\mathcal{H}}\). Applications to symmetric operators, symmetric operators perturbed by a relatively bounded dissipative operator and more singular differential operators are discussed. Finally, we investigate the stability of the numerical range of the different dissipative extensions.     

{{C^m}} Positive Eigenvectors for Linear Operators Arising in the Computation of Hausdorff Dimension

We consider a broad class of linear Perron–Frobenius operators \({\Lambda:X \rightarrow X}\), where \({X}\) is a real Banach space of \({C^m}\) functions. We prove the existence of a strictly positive \({C^m}\) eigenvector \({v}\) with eigenvalue \({r=r(\Lambda) =}\) the spectral radius of \({\Lambda}\). We prove (see Theorem 6.5 in Sect. 6 of this paper) that \({r(\Lambda)}\) is an algebraically simple eigenvalue and that, if \({\sigma(\Lambda)}\) denotes the spectrum of the complexification of \({\Lambda,\sigma(\Lambda) \backslash \{r(\Lambda)\}\subseteq \{\zeta \in \mathbb{C} \big| |\zeta| \le r_*\}}\), where \({r_* < r(\Lambda)}\). Furthermore, if \({u \in X}\) is any strictly positive function, \({(\frac 1r \Lambda)^k(u) \rightarrow s_u v}\) as \({k \rightarrow \infty}\), where \({s_u > 0}\) and convergence is in the norm topology on \({X}\). In applications to the computation of Hausdorff dimension, one is given a parametrized family \({\Lambda_s,s > s_*}\), of such operators and one wants to determine the (unique) value \({s_0}\) such that \({r(\Lambda_{s_0})=1}\). In another paper (Falk and Nussbaum in C\({^{\rm m}}\) Eigenfunctions of Perron–Frobenius operators and a new approach to numerical computation of Hausdorff dimension, submitted) we prove that explicit estimates on the partial derivatives of the positive eigenvector \({v_s}\) of \({\Lambda_s}\) can be obtained and that this information can be used to give rigorous, sharp upper and lower bounds for \({s_0}\).     

Sign-changing solutions for supercritical elliptic problems in domains with small holes

Let \(\mathcal {D}\) be a bounded, smooth domain in \(\mathbb {R}^N\) , N ≥ 3, \(P\in \mathcal {D}\) . We consider the boundary value problem in \(\Omega = \mathcal {D} \setminus B_\delta(P)\) ,

$\begin{aligned}\Delta u + |u|^{p-1} u = 0\, \quad in\, \Omega,\\u = 0\quad on\, \partial\Omega,\end{aligned}$

with p supercritical, namely \(p > \frac{N+2}{N-2}\) . Given any positive integer m, we find a sequence \(p_1 < p_2 < p_3 < \cdots , \quad with \lim_{k\to+\infty} p_k = +\infty \), such that if p is given, with p ≠ p _j for all j, then for all δ > 0 sufficiently small, this problem has a sign-changing solution which has exactly m + 1 nodal domains.

     

Gauge functions,Eikonal equations and Bôcher’s theorem on stratified Lie groups

Let \({\mathcal{L} = \sum_{i=1}^m X_i^2}\) be a real sub-Laplacian on a Carnot group \({\mathbb{G}}\) and denote by \({\nabla_\mathcal{L} = (X_1,\ldots,X_m)}\) the intrinsic gradient related to \({\mathcal{L}}\). Our aim in this present paper is to analyze some features of the \({\mathcal{L}}\)-gauge functions on \({\mathbb{G}}\), i.e., the homogeneous functions d such that \({\mathcal{L}(d^\gamma) = 0}\) in \({\mathbb{G} \setminus \{0\}}\) , for some \({\gamma \in \mathbb{R} \setminus \{0\}}\). We consider the relation of \({\mathcal{L}}\)-gauge functions with: the \({\mathcal{L}}\)-Eikonal equation \({|\nabla_\mathcal{L} u| = 1}\) in \({\mathbb{G}}\); the Mean Value Formulas for the \({\mathcal{L}}\)-harmonic functions; the fundamental solution for \({\mathcal{L}}\); the Bôcher-type theorems for nonnegative \({\mathcal{L}}\)-harmonic functions in “punctured” open sets \({\dot \Omega:= \Omega \setminus \{x_0\}}\).     

On estimation of delay location

Assume that we observe a stationary Gaussian process X(t), \({t \in [-r, T]}\) , which satisfies the affine stochastic delay differential equation

$d X(t) = \int\limits_{[-r,0]}X(t+u)\, a_\vartheta (du)\,dt +dW(t), \quad t\ge 0,$

where W(t), t ≥ 0, is a standard Wiener process independent of X(t), \({t\in [-r, 0]}\) , and \({a_\vartheta}\) is a finite signed measure on [?r, 0], \({\vartheta\in\Theta}\) . The parameter \({\vartheta}\) is unknown and has to be estimated based on the observation. In this paper we consider the case where \({\Theta=(\vartheta_0,\vartheta_1)}\) , \({-\infty\,<\,\vartheta_0 <0 \,<\,\vartheta_1\,<\,\infty}\) , and the measures \({a_\vartheta}\) are of the form

$a_\vartheta = a+b_\vartheta-b,$

where a and b are finite signed measure on [?r, 0] and \({b_\vartheta}\) is the translate of b by \({\vartheta}\) . We study the limit behaviour of the normalized likelihoods

$Z_{T,\vartheta}(u) = \frac{dP_T^{\vartheta+\delta_T u}}{dP_T^\vartheta}$

as T→ ∞, where \({P_T^\vartheta}\) is the distribution of the observation if the true value of the parameter is \({\vartheta}\) . A necessary and sufficient condition for the existence of a rescaling function δ _T such that \({Z_{T,\vartheta}(u)}\) converges in distribution to an appropriate nondegenerate limiting function \({Z_{\vartheta}(u)}\) is found. It turns out that then the limiting function \({Z_{\vartheta}(u)}\) is of the form

$Z_\vartheta(u)=\exp\left(B^H(u) - E[B^H(u)]^2/2\right),$

where \({H\in[1/2,1]}\) and B ^H (u), \({u\in\mathbb{R}}\) , is a fractional Brownian motion with index H, and δ _T  = T ^?1/(2H) ?(T) with a slowly varying function ?. Every \({H\in[1/2,1]}\) may occur in this framework. As a consequence, the asymptotic behaviour of maximum likelihood and Bayes estimators is found.

     

Blow-up phenomena for a nonlocal <Emphasis Type="Italic">p</Emphasis>-Laplace equation with Neumann boundary conditions

This paper is concerned with the blow-up of solutions to the following nonlocal p-Laplace equation:

$$u_t-\mathrm{div}(|\nabla{u}|^{p-2}\nabla{u})=|u|^{q-1}u-\frac{1}{|\Omega|} \int\limits_\Omega{|u|^{q-1}u}dx,\quad x\in\Omega,\quad 0 < t < T,$$

under homogeneous Neumann boundary conditions in a bounded smooth domain \({\Omega\subset\mathrm{R}^N}\). For all \({p > 2, q > p-1}\), a blow-up result for the solutions to the above equation with positive initial energy is established. This result improves a recent result by Qu and Liang (Abstr Appl Anal 3:551–552, 2013) which asserts the blow-up of solutions for \({p-1 < q\leq\frac{Np}{(N-p)_+}-1}\).

     

Symmetry for a general class of overdetermined elliptic problems

Let \({\Omega}\) a bounded domain in \({\mathbb{R} ^N }\), and let \({u\in C^1 (\overline{\Omega})}\) a weak solution of the following overdetermined BVP: \({-\nabla (g(|\nabla u|)|\nabla u|^{-1}\nabla u)=f(|x|,u)}\), \({ u > 0 }\) in \({\Omega }\) and \({u=0, \ |\nabla u(x)|=\lambda (|x|)}\) on \({\partial \Omega }\), where \({g\in C([0,+\infty)\cap C^1 ((0,+\infty ) ) }\) with \({g(0)=0}\), \({g'(t) > 0}\) for \({t > 0}\), \({f\in C([0,+\infty ) \times [0, +\infty ) )}\), f is nonincreasing in \({|x|}\), \({\lambda \in C([0, +\infty )) }\) and \({\lambda }\) is positive and nondecreasing. We show that \({\Omega }\) is a ball and u satisfies some “local” kind of symmetry. The proof is based on the method of continuous Steiner symmetrization.     

Existence of maximal solutions for some very singular nonlinear fractional diffusion equations in 1D

We consider nonlinear parabolic equations involving fractional diffusion of the form \({\partial_t u + {(-\Delta)}^{s} \Phi(u)= 0,}\) with \({0 < s < 1}\), and solve an open problem concerning the existence of solutions for very singular nonlinearities \({\Phi}\) in power form, precisely \({\Phi'(u)=c\,u^{-(n+1)}}\) for some \({0 < n < 1}\). We also include the logarithmic diffusion equation \({\partial_t u + {(-\Delta)}^{s} \log(u)= 0}\), which appears as the case \({n=0}\). We consider the Cauchy problem with nonnegative and integrable data \({u_0(x)}\) in one space dimension, since the same problem in higher dimensions admits no nontrivial solutions according to recent results of the author and collaborators. The limit solutions we construct are unique, conserve mass, and are in fact maximal solutions of the problem. We also construct self-similar solutions of Barenblatt type, which are used as a cornerstone in the existence theory, and we prove that they are asymptotic attractors (as \({t\to\infty}\)) of the solutions with general integrable data. A new comparison principle is introduced.     

On the equation $${{\rm det}\,\nabla{u}=f}$$ with no sign hypothesis

We prove existence of \({u\in C^{k}(\overline{\Omega};\mathbb{R}^{n})}\) satisfying

$\left\{\begin{array}{ll} det\nabla u(x) =f(x) \, x\in \Omega\\ u(x) =x \quad\quad\quad\quad x\in\partial\Omega\end{array}\right.$

where k ≥ 1 is an integer, \({\Omega}\) is a bounded smooth domain and \({f\in C^{k}(\overline{\Omega}) }\) satisfies

$\int\limits_{\Omega}f(x) dx={\rm meas} \Omega$

with no sign hypothesis on f.

     

Dirac–Krein Systems on Star Graphs

Suppose that \({\mathcal {M}}\) is a countably decomposable type II\({_1}\) von Neumann algebra and \({\mathcal {A}}\) is a separable, non-nuclear, unital C\({^*}\)-algebra. We show that, if \({\mathcal {M}}\) has Property \({\Gamma}\), then the similarity degree of \({\mathcal {M}}\) is less than or equal to 5. If \({\mathcal {A}}\) has Property c\({^*}\)-\({\Gamma}\), then the similarity degree of \({\mathcal {A}}\) is equal to 3. In particular, the similarity degree of a \({\mathcal {Z}}\)-stable, separable, non-nuclear, unital C\({^*}\)-algebra is equal to 3.     

On Ambrosetti–Malchiodi–Ni conjecture on two-dimensional smooth bounded domains

We consider the problem

$$\begin{aligned} \epsilon ^2 \Delta u-V(y)u+u^p\,=\,0,\quad u>0\quad \text{ in }\quad \Omega , \quad \frac{\partial u}{\partial \nu }\,=\,0\quad \text{ on }\quad \partial \Omega , \end{aligned}$$

where \(\Omega \) is a bounded domain in \({\mathbb {R}}^2\) with smooth boundary, the exponent p is greater than 1, \(\epsilon >0\) is a small parameter, V is a uniformly positive, smooth potential on \(\bar{\Omega }\), and \(\nu \) denotes the outward unit normal of \(\partial \Omega \). Let \(\Gamma \) be a curve intersecting orthogonally \(\partial \Omega \) at exactly two points and dividing \(\Omega \) into two parts. Moreover, \(\Gamma \) satisfies stationary and non-degeneracy conditions with respect to the functional \(\int _{\Gamma }V^{\sigma }\), where \(\sigma =\frac{p+1}{p-1}-\frac{1}{2}\). We prove the existence of a solution \(u_\epsilon \) concentrating along the whole of \(\Gamma \), exponentially small in \(\epsilon \) at any fixed distance from it, provided that \(\epsilon \) is small and away from certain critical numbers. In particular, this establishes the validity of the two dimensional case of a conjecture by Ambrosetti et al. (Indiana Univ Math J 53(2), 297–329, 2004).

     

A semilinear parabolic problem with singular term at the boundary

In this paper, we deal with a class of semilinear parabolic problems related to a Hardy inequality with singular weight at the boundary.

More precisely, we consider the problem

$$\left\{\begin{array}{l@{\quad}l}u_t-\Delta u=\lambda \frac{u^p}{d^2}&\text{ in }\,\Omega_{T}\equiv\Omega \times (0,T), \\u>0 &\text{ in }\,{\Omega_T}, \\u(x,0)=u_0(x)>0 &\text{ in }\,\Omega, \\u=0 &\text{ on }\partial \Omega \times (0,T),\end{array}\right.$$

(P)

where Ω is a bounded regular domain of \({\mathbbm{R}^N}\), \({d(x)=\text{dist}(x,\partial\Omega)}\), \({p > 0}\), and \({\lambda > 0}\) is a positive constant.

We prove that

1. 1.
   If \({0 < p < 1}\), then (P) has no positive very weak solution.
    
2. 2.
   If \({p=1}\), then (P) has a positive very weak solution under additional hypotheses on \({\lambda}\) and \({u_0}\).
    
3. 3.
   If \({p > 1}\), then, for all \({\lambda > 0}\), the problem (P) has a positive very weak solution under suitable hypothesis on \({u_0}\).
    

Moreover, we consider also the concave–convex-related case.

     

Continuity of the first eigenvalue for a family of degenerate eigenvalue problems

For each \({\alpha\in[0,2)}\) we consider the eigenvalue problem \({-{\rm div}(|x|^\alpha \nabla u)=\lambda u}\) in a bounded domain \({\Omega\subset \mathbb{R}^N}\) (\({N\geq 2}\)) with smooth boundary and \({0\in \Omega}\) subject to the homogeneous Dirichlet boundary condition. Denote by \({\lambda_1(\alpha)}\) the first eigenvalue of this problem. Using \({\Gamma}\)-convergence arguments we prove the continuity of the function \({\lambda_1}\) with respect to \({\alpha}\) on the interval \({[0,2)}\).     

On one-dimensional superlinear indefinite problems involving the $$\phi $$-Laplacian

Let \(\Omega := ( a,b ) \subset \mathbb {R}\), \(m\in L^{1} ( \Omega ) \) and \(\phi :\mathbb {R\rightarrow R}\) be an odd increasing homeomorphism. We consider the existence of positive solutions for problems of the form

$$\begin{aligned} \left\{ \begin{array} [c]{ll} -\phi ( u^{\prime } ) ^{\prime }=m ( x ) f ( u) &{}\quad \text {in } \Omega ,\\ u=0 &{}\quad \text {on } \partial \Omega , \end{array} \right. \end{aligned}$$

where \(f: [ 0,\infty ) \rightarrow [ 0,\infty ) \) is a continuous function which is, roughly speaking, superlinear with respect to \(\phi \). Our approach combines the Guo-Krasnoselski? fixed-point theorem with some estimates on related nonlinear problems. We mention that our results are new even in the case \(m\ge 0\).

     

Solutions of perturbed Schrödinger equations with critical nonlinearity

We consider the perturbed Schrödinger equation

$\left\{\begin{array}{ll}{- \varepsilon ^2 \Delta u + V(x)u = P(x)|u|^{p - 2} u + k(x)|u|^{2* - 2} u} &; {\text{for}}\, x \in {\mathbb{R}}^N\\ \qquad \qquad \quad {u(x) \rightarrow 0} &; \text{as}\, {|x| \rightarrow \infty} \end{array} \right.$

where \(N\geq 3, \ 2^*=2N/(N-2)\) is the Sobolev critical exponent, \(p\in (2, 2^*)\) , P(x) and K(x) are bounded positive functions. Under proper conditions on V we show that it has at least one positive solution provided that \(\varepsilon\leq{\mathcal{E}}\) ; for any \(m\in{\mathbb{N}}\) , it has m pairs of solutions if \(\varepsilon\leq{\mathcal{E}}_{m}\) ; and suppose there exists an orthogonal involution \(\tau:{\mathbb{R}}^{N}\to{\mathbb{R}}^{N}\) such that V(x), P(x) and K(x) are τ -invariant, then it has at least one pair of solutions which change sign exactly once provided that \(\varepsilon\leq{\mathcal{E}}\) , where \({\mathcal{E}}\) and \({\mathcal{E}}_{m}\) are sufficiently small positive numbers. Moreover, these solutions \(u_\varepsilon\to 0\) in \(H^1({\mathbb{R}}^N)\) as \(\varepsilon\to 0\) .