The fractional Neumann and Robin type boundary conditions for the regional fractional <Emphasis Type="Italic">p</Emphasis>-Laplacian

Let \({p \in (1,\infty)}\), \({s \in (0,1)}\) and \({\Omega \subset {\mathbb{R}^{N}}}\) a bounded open set with boundary \({\partial\Omega}\) of class C ^1,1. In the first part of the article we prove an integration by parts formula for the fractional p-Laplace operator \({(-\Delta)_{p}^{s}}\) defined on \({\Omega \subset {\mathbb{R}^{N}}}\) and acting on functions that do not necessarily vanish at the boundary \({\partial\Omega}\). In the second part of the article we use the above mentioned integration by parts formula to clarify the fractional Neumann and Robin boundary conditions associated with the fractional p-Laplacian on open sets.     

Interpolation of sum and intersection spaces of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>-type and applications to the Stokes problem in general unbounded domains

In a general unbounded uniform C ²-domain \({\Omega \subset \mathbb{R}^n, n \geq 3}\) , and \({1\leq q\leq \infty}\) consider the spaces \({\tilde{L}^q(\Omega)}\) defined by \({\tilde{L^q}(\Omega) := \left\{\begin{array}{ll}L^q(\Omega)+L^2(\Omega),\quad q < 2, \\ L^q(\Omega)\cap L^2(\Omega),\quad q\geq 2, \end{array}\right.}\) and corresponding subspaces of solenoidal vector fields, \({\tilde{L}^q_\sigma(\Omega)}\) . By studying the complex and real interpolation spaces of these we derive embedding properties for fractional order spaces related to the Stokes problem and L ^p ? L ^q -type estimates for the corresponding semigroup.     

Homoclinic Solutions for <Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">t</Emphasis>)-Laplacian–Hamiltonian Systems Without Coercive Conditions

In this paper, we study the existence and multiplicity of homoclinic solutions for the following second-order p(t)-Laplacian–Hamiltonian systems

$$\frac{{\rm d}}{{\rm d}t}(|\dot{u}(t)|^{p(t)-2}\dot{u}(t))-a(t)|u(t)|^{p(t)-2}u(t)+\nabla W(t,u(t))=0,$$

where \({t \in \mathbb{R}}\), \({u \in \mathbb{R}^n}\), \({p \in C(\mathbb{R},\mathbb{R})}\) with p(t) > 1, \({a \in C(\mathbb{R},\mathbb{R})}\), \({W\in C^1(\mathbb{R}\times\mathbb{R}^n,\mathbb{R})}\) and \({\nabla W(t,u)}\) is the gradient of W(t, u) in u. The point is that, assuming that a(t) is bounded in the sense that there are constants \({0<\tau_1<\tau_2<\infty}\) such that \({\tau_1\leq a(t)\leq \tau_2 }\) for all \({t \in \mathbb{R}}\) and W(t, u) is of super-p(t) growth or sub-p(t) growth as \({|u|\rightarrow \infty}\), we provide two new criteria to ensure the existence and multiplicity of homoclinic solutions, respectively. Recent results in the literature are extended and significantly improved.

     

Hardy Inequalities for Finsler <Emphasis Type="Italic">p</Emphasis>-Laplacian in the Exterior Domain

The Finsler p-Laplacian is the class of nonlinear differential operators given by

$$\begin{aligned} \Delta _{H,p}u:= \text {div}(H(\nabla u)^{p-1}\nabla _{\eta } H(\nabla u)) \end{aligned}$$

where \(1<p<\infty \) and \(H:\mathbf {R}^N\rightarrow [0,\infty )\) is in \(C^2(\mathbf {R}^N\backslash \{0\})\) and is positively homogeneous of degree 1. Under some additional constraints on H, we derive the Hardy inequality for Finsler p-Laplacian in exterior domain for \(1<p\le N\). We also provide an improved version of Hardy inequality for the case \(p=2\).

     

Ground states of a nonlinear curl-curl problem in cylindrically symmetric media

We consider the nonlinear curl-curl problem \({\nabla\times\nabla\times U + V(x) U= \Gamma(x)|U|^{p-1}U}\) in \({\mathbb{R}^3}\) related to the Kerr nonlinear Maxwell equations for fully localized monochromatic fields. We search for solutions as minimizers (ground states) of the corresponding energy functional defined on subspaces (defocusing case) or natural constraints (focusing case) of \({H({\rm curl};\mathbb{R}^3)}\). Under a cylindrical symmetry assumption corresponding to a photonic fiber geometry on the functions V and \({\Gamma}\) the variational problem can be posed in a symmetric subspace of \({H({\rm curl};\mathbb{R}^3)}\). For a defocusing case \({{\rm sup} \Gamma < 0}\) with large negative values of \({\Gamma}\) at infinity we obtain ground states by the direct minimization method. For the focusing case \({{\rm inf} \Gamma > 0}\) the concentration compactness principle produces ground states under the assumption that zero lies outside the spectrum of the linear operator \({\nabla \times \nabla \times +V(x)}\). Examples of cylindrically symmetric functions V are provided for which this holds.     

Asymptotic behavior of the p-torsion functions as p goes to 1

Let \({\Omega}\) be a Lipschitz bounded domain of \({\mathbb{R}^N}\), \({N\geq2}\), and let \({u_p\in W_0^{1,p}(\Omega)}\) denote the p-torsion function of \({\Omega}\), p > 1. It is observed that the value 1 for the Cheeger constant \({h(\Omega)}\) is threshold with respect to the asymptotic behavior of u_p, as \({p\rightarrow 1^+}\), in the following sense: when \({h(\Omega) > 1}\), one has \({\lim_{p\rightarrow 1^+}\left\|u_{p}\right\| _{L^\infty(\Omega)}=0}\), and when \({h(\Omega) < 1}\), one has \({\lim_{p\rightarrow 1^+}\left\|u_p\right\| _{L^\infty(\Omega)}=\infty}\). In the case \({h(\Omega)=1}\), it is proved that \({\limsup_{p\rightarrow1^+}\left\|u_p\right\|_{L^\infty(\Omega)}<\infty}\). For a radial annulus \({\Omega_{a,b}}\), with inner radius a and outer radius b, it is proved that \({\lim_{p\rightarrow 1^+}\left\|u_p\right\| _{L^\infty(\Omega_{a,b})}=0}\) when \({h(\Omega_{a,b})=1}\).     

Maximum Principle for H-Surfaces in the Unit Cone and Dirichlet’s Problem for their Equation in Central Projection

In the unit cone\({\mathcal{C} := \{(x, y, z)} \in {\mathbb R}^{3} : {x}^{2} + {y}^{2} < {z}^{2}, {z} > {0}\}\) we establish a geometric maximum principle for H-surfaces, where its mean curvature \({H = H(x, y, z)}\) is optimally bounded. Consequently, these surfaces cannot touch the conical boundary \({\partial \mathcal{C}}\) at interior points and have to approach \({\partial \mathcal{C}}\) transversally. By a nonlinear continuity method, we then solve the Dirichlet problem of the H-surface equation in central projection for Jordan-domains \({\Omega}\) which are strictly convex in the following sense: On its whole boundary \({\partial \mathcal{C}(\Omega)}\) their associate cone \({\mathcal{C}(\Omega) := \{(rx, ry, r) \in {\mathbb R}^{3} : (x, y) \in \Omega, r \in (0,+\infty)}\}\) admits rotated unit cones \({O \circ \mathcal{C}}\) as solids of support, where \({O \in {\mathbb R}^{3\times3}}\) represents a rotation in the Euclidean space. Thus we construct the unique H-surface with one-to-one central projection onto these domains \({\Omega}\) bounding a given Jordan-contour \({\Gamma \subset \mathcal{C} \backslash \{0\}}\) with one-toone central projection.     

Heat kernel approach for sup-norm bounds for cusp forms of integral and half-integral weight

In this article, using the heat kernel approach from Bouche (Asymptotic results for Hermitian line bundles over complex manifolds: The heat kernel approach, Higher-dimensional complex varieties, pp 67–81, de Gruyter, Berlin, 1996), we derive sup-norm bounds for cusp forms of integral and half-integral weight. Let \({\Gamma\subset \mathrm{PSL}_{2}(\mathbb{R})}\) be a cocompact Fuchsian subgroup of first kind. For \({k \in \frac{1}{2} \mathbb{Z}}\) (or \({k \in 2\mathbb{Z}}\)), let \({S^{k}_{\nu}(\Gamma)}\) denote the complex vector space of cusp forms of weight-k and nebentypus \({\nu^{2k}}\) (\({\nu^{k\slash 2}}\), if \({k \in 2\mathbb{Z}}\)) with respect to \({\Gamma}\), where \({\nu}\) is a unitary character. Let \({\lbrace f_{1},\ldots,f_{j_{k}} \rbrace}\) denote an orthonormal basis of \({S^{k}_{\nu}(\Gamma)}\). In this article, we show that as \({k \rightarrow \infty,}\) the sup-norm for \({\sum_{i=1}^{j_{k}}y^{k}|f_{i}(z)|^{2}}\) is bounded by O(k), where the implied constant is independent of \({\Gamma}\). Furthermore, using results from Berman (Math. Z. 248:325–344, 2004), we extend these results to the case when \({\Gamma}\) is cofinite.     

Some estimates for the Schrödinger type operators with nonnegative potentials

In this paper we consider the Schrödinger operator ?Δ + V on \({\mathbb R^d}\), where the nonnegative potential V belongs to the reverse Hölder class \({B_{q_{_1}}}\) for some \({q_{_1}\geq \frac{d}{2}}\) with d ≥ 3. Let \({H^1_L(\mathbb R^d)}\) denote the Hardy space related to the Schrödinger operator L = ?Δ + V and \({BMO_L(\mathbb R^d)}\) be the dual space of \({H^1_L(\mathbb R^d)}\). We show that the Schrödinger type operator \({\nabla(-\Delta +V)^{-\beta}}\) is bounded from \({H^1_L(\mathbb R^d)}\) into \({L^p(\mathbb R^d)}\) for \({p=\frac{d}{d-(2\beta-1)}}\) with \({ \frac{1}{2}<\beta<\frac{3}{2} }\) and that it is also bounded from \({L^p(\mathbb R^d)}\) into \({BMO_L(\mathbb R^d)}\) for \({p=\frac{d}{2\beta-1}}\) with \({ \frac{1}{2}<\beta< 2}\).     

Lattice homomorphisms between Sobolev spaces

We show in Theorem 4.4 that every vector lattice homomorphism T from \({\mathsf{W}^{1,p}_0(\Omega_1)}\) into \({\mathsf{W}^{1,q}(\Omega_2)}\) for \({p,q\in (1,\infty)}\) and open sets \({\Omega_1,\Omega_2\subset\mathbb{R}^N}\) has a representation of the form \({T\mathsf{u}=(\mathsf{u}\circ\xi)g}\) (Cap _q -quasi everywhere on Ω₂) with mappings ξ : Ω₂ → Ω₁ and g : Ω₂ → [0, ∞). This representation follows as an application of an abstract and more general representation theorem (Theorem 3.5). Other applications are also given.     

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.

     

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.     

Conformal embeddings of affine vertex algebras in minimal <Emphasis Type="Italic">W</Emphasis>-algebras II: decompositions

We present methods for computing the explicit decomposition of the minimal simple affine W-algebra \({W_k(\mathfrak{g}, \theta)}\) as a module for its maximal affine subalgebra \({\mathscr{V}_k(\mathfrak{g}^{\natural})}\) at a conformal level k, that is, whenever the Virasoro vectors of \({W_k(\mathfrak{g}, \theta)}\) and \({\mathscr{V}_k(\mathfrak{g}^\natural)}\) coincide. A particular emphasis is given on the application of affine fusion rules to the determination of branching rules. In almost all cases when \({\mathfrak{g}^{\natural}}\) is a semisimple Lie algebra, we show that, for a suitable conformal level k, \({W_k(\mathfrak{g}, \theta)}\) is isomorphic to an extension of \({\mathscr{V}_k(\mathfrak{g}^{\natural})}\) by its simple module. We are able to prove that in certain cases \({W_k(\mathfrak{g}, \theta)}\) is a simple current extension of \({\mathscr{V}_k(\mathfrak{g}^{\natural})}\). In order to analyze more complicated non simple current extensions at conformal levels, we present an explicit realization of the simple W-algebra \({W_{k}(\mathit{sl}(4), \theta)}\) at k = ?8/3. We prove, as conjectured in [3], that \({W_{k}(\mathit{sl}(4), \theta)}\) is isomorphic to the vertex algebra \({\mathscr{R}^{(3)}}\), and construct infinitely many singular vectors using screening operators. We also construct a new family of simple current modules for the vertex algebra \({V_k (\mathit{sl}(n))}\) at certain admissible levels and for \({V_k (\mathit{sl}(m \vert n)), m\ne n, m,n\geq 1}\) at arbitrary levels.     

Neostability-properties of Fraïssé limits of 2-nilpotent groups of exponent {p > 2}

Let \({L(n)}\) be the language of group theory with n additional new constant symbols \({c_1,\ldots,c_n}\). In \({L(n)}\) we consider the class \({{\mathbb{K}}(n)}\) of all finite groups G of exponent \({p > 2}\), where \({G'\subseteq\langle c_1^G,\ldots,c_n^G\rangle \subseteq Z(G)}\) and \({c_1^G,\ldots,c_n^G}\) are linearly independent. Using amalgamation we show the existence of Fraïssé limits \({D(n)}\) of \({{\mathbb{K}}(n)}\). \({D(1)}\) is Felgner’s extra special p-group. The elementary theories of the \({D(n)}\) are supersimple of SU-rank 1. They have the independence property.     

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.

     

On a two-phase free boundary condition for <Emphasis Type="Italic">p</Emphasis>-harmonic measures

Let \({\Omega^i\subset {\bf R}^n, i\in\{1,2\}}\) , be two (δ, r ₀)-Reifenberg flat domains, for some \({0 < \delta < \hat \delta}\) and r ₀ > 0, assume \({\Omega^1\cap\Omega^2=\emptyset}\) and that, for some \({w\in {\bf R}^n}\) and some 0 < r, \({w\in\partial\Omega^1\cap\partial\Omega^2, \partial\Omega^1\cap B(w,2r)=\partial\Omega^2\cap B(w,2r)}\) . Let p, 1 < p < ∞, be given and let u ⁱ , \({i\in\{1,2\}}\) , denote a non-negative p-harmonic function in Ω ⁱ , assume that u ⁱ , \({i\in\{1,2\}}\), is continuous in \({\bar\Omega^i\cap B(w,2r) }\) and that u ⁱ  = 0 on \({\partial\Omega^i\cap B(w,2r)}\) . Extend u ⁱ to B(w, 2r) by defining \({u^i\equiv 0}\) on \({B(w,2r) {\setminus} \Omega^i}\). Then there exists a unique finite positive Borel measure μ ⁱ , \({i\in\{1,2\}}\) , on R ⁿ , with support in \({\partial\Omega^i\cap B(w,2r)}\) , such that if \({\phi \in C_0^\infty (B(w,2r))}\) , then

$\int\limits_{\mathbf R^n} \,|\nabla u^i|^{ p-2} \,\langle \nabla u^i, \,\nabla \phi \rangle \,dx =- \int\limits_{\mathbf R^n} \,\phi \,d \mu^i.$

Let \({\Delta(w,2r)=\partial\Omega^1\cap B(w,2r)=\partial\Omega^2\cap B(w,2r)}\) . The main result proved in this paper is the following. Assume that μ ² is absolutely continuous with respect to μ ¹ on Δ(w, 2r), d μ ² = kd μ ¹ for μ ¹-almost every point in Δ(w, 2r) and that \({\log k\in VMO(\Delta(w,r),\mu^1)}\) . Then there exists \({\tilde \delta = \tilde \delta(p,n) > 0}\) , \({\tilde \delta < \hat \delta}\) , such that if \({\delta\leq\tilde\delta}\) , then Δ(w, r/2) is Reifenberg flat with vanishing constant. Moreover, the special case p = 2, i.e., the linear case and the corresponding problem for harmonic measures, has previously been studied in Kenig and Toro (J Reine Angew Math 596:1–44, 2006).

     

Stable equilibria of a singularly perturbed reaction–diffusion equation when the roots of the degenerate equation contact or intersect along a non-smooth hypersurface

We use the variational concept of \({\Gamma}\)-convergence to prove existence, stability and exhibit the geometric structure of four families of stationary solutions to the singularly perturbed parabolic equation \({u_t=\epsilon^2 {\rm div}(k\nabla u)+f(u,x)}\), for \({(t,x)\in \mathbb{R}^+\times\Omega}\), where \({\Omega\subset\mathbb{R}^n}\), \({n\geq 1}\), supplied with no-flux boundary condition. The novelty here lies in the fact that the roots of the bistable function f are not isolated, meaning that the graphs of its roots are allowed to have contact or intersect each other along a Lipschitz-continuous (n ? 1)-dimensional hypersurface \({\gamma \subset \Omega}\); across this hypersurface, the stable equilibria may have corners. The case of intersecting roots stems from the phenomenon known as exchange of stability which is characterized by \({f(\cdot,x)}\) having only two roots.     

Hecke algebras for {{\rm GL}_n} over local fields

We study the local Hecke algebra \({\mathcal{H}_{G}(K)}\) for \({G = {\rm GL}_{n}}\) and K a non-archimedean local field of characteristic zero. We show that for \({G = {\rm GL}_{2}}\) and any two such fields K and L, there is a Morita equivalence \({\mathcal{H}_{G}(K) \sim_{M} \mathcal{H}_{G}(L)}\), by using the Bernstein decomposition of the Hecke algebra and determining the intertwining algebras that yield the Bernstein blocks up to Morita equivalence. By contrast, we prove that for \({G = {\rm GL}_{n}}\), there is an algebra isomorphism \({\mathcal{H}_{G}(K) \cong \mathcal{H}_{G}(L)}\) which is an isometry for the induced \({L^1}\)-norm if and only if there is a field isomorphism \({K \cong L}\).     

The Katchalski–Lewis transversal problem for regular polygons

If every k-membered subfamily of a family of plane convex bodies has a line transversal, then we say that this family has property T(k). We say that a family \({\mathcal{F}}\) has property \({T-m}\), if there exists a subfamily \({\mathcal{G} \subset \mathcal{F}}\) with \({|\mathcal{F} - \mathcal{G}| \le m}\) admitting a line transversal. Heppes [7] posed the problem whether there exists a convex body K in the plane such that if \({\mathcal{F}}\) is a finite T(3)-family of disjoint translates of K, then m = 3 is the smallest value for which \({\mathcal{F}}\) has property \({T-m}\). In this paper, we study this open problem in terms of finite T(3)-families of pairwise disjoint translates of a regular 2n-gon \({(n \ge 5)}\). We find out that, for \({5 \le n \le 34}\), the family has property \({T - 3}\) ; for \({n \ge 35}\), the family has property \({T - 2}\).     

$${\sigma_2}$$-Energy as a polyconvex functional on a space of self-maps of annuli in the multi-dimensional calculus of variations

Let \({\mathbb{X} \subset \mathbb {R}^n}\) be a bounded Lipschitz domain and consider the energy functional

$${{\mathbb F}_{\sigma_2}}[u; \mathbb{X}] := \int_\mathbb{X} {\mathbf F}(\nabla u) \, dx,$$

over the space of admissible maps

$${{\mathcal {A}_\varphi}(\mathbb{X}) :=\{u \in W^{1,4}(\mathbb{X}, {\mathbb{R}^n}) : {\rm det}\, \nabla u > 0\, {\rm for}\, {\mathcal {L}^n}{\rm -a.e. in}\, \mathbb{X}, u|_{\partial \mathbb{X}} =\varphi \}},$$

where the integrand \({{\mathbf F}\colon \mathbb M_{n\times n}\to \mathbb{R}}\) is quasiconvex and sufficiently regular. Here our attention is paid to the prototypical case when \({{\mathbf F}(\xi):=\frac{1}{2}\sigma_2(\xi)+\Phi(\det\xi)}\). The aim of this paper is to discuss the question of multiplicity versus uniqueness for extremals and strong local minimizers of \({\mathbb F_{\sigma_2}}\) and the relation it bares to the domain topology. In contrast, for constructing explicitly and directly solutions to the system of Euler–Lagrange equations associated to \({{\mathbb F}_{\sigma_2}}\), we use a topological class of maps referred to as generalised twists and relate the problem to extremising an associated energy on the compact Lie group \({\mathbf {SO}(n)}\). The main result is a surprising discrepancy between even and odd dimensions. In even dimensions the latter system of equations admits infinitely many smooth solutions amongst such maps whereas in odd dimensions this number reduces to one.