Nonlinear Schrödinger equations near an infinite well potential

The paper deals with standing wave solutions of the dimensionless nonlinear Schrödinger equation where the potential \(V_\lambda :\mathbb {R}^N\rightarrow \mathbb {R}\) is close to an infinite well potential \(V_\infty :\mathbb {R}^N\rightarrow \mathbb {R}\) , i. e. \(V_\infty =\infty \) on an exterior domain \(\mathbb {R}^N\setminus \Omega \) , \(V_\infty |_\Omega \in L^\infty (\Omega )\) , and \(V_\lambda \rightarrow V_\infty \) as \(\lambda \rightarrow \infty \) in a sense to be made precise. The nonlinearity may be of Gross–Pitaevskii type. A standing wave solution of \((NLS_\lambda )\) with \(\lambda =\infty \) vanishes on \(\mathbb {R}^N\setminus \Omega \) and satisfies Dirichlet boundary conditions, hence it solves We investigate when a standing wave solution \(\Phi _\infty \) of the infinite well potential \((NLS_\infty )\) gives rise to nearby solutions \(\Phi _\lambda \) of the finite well potential \((NLS_\lambda )\) with \(\lambda \gg 1\) large. Considering \((NLS_\infty )\) as a singular limit of \((NLS_\lambda )\) we prove a kind of singular continuation type results.     

Mean Squared Error Minimization for Inverse Moment Problems

We consider the problem of approximating the unknown density \(u\in L^2(\Omega ,\lambda )\) of a measure \(\mu \) on \(\Omega \subset \mathbb {R}^n\) , absolutely continuous with respect to some given reference measure \(\lambda \) , only from the knowledge of finitely many moments of \(\mu \) . Given \(d\in \mathbb {N}\) and moments of order \(d\) , we provide a polynomial \(p_d\) which minimizes the mean square error \(\int (u-p)^2d\lambda \) over all polynomials \(p\) of degree at most \(d\) . If there is no additional requirement, \(p_d\) is obtained as solution of a linear system. In addition, if \(p_d\) is expressed in the basis of polynomials that are orthonormal with respect to \(\lambda \) , its vector of coefficients is just the vector of given moments and no computation is needed. Moreover \(p_d\rightarrow u\) in \(L^2(\Omega ,\lambda )\) as \(d\rightarrow \infty \) . In general nonnegativity of \(p_d\) is not guaranteed even though \(u\) is nonnegative. However, with this additional nonnegativity requirement one obtains analogous results but computing \(p_d\ge 0\) that minimizes \(\int (u-p)^2d\lambda \) now requires solving an appropriate semidefinite program. We have tested the approach on some applications arising from the reconstruction of geometrical objects and the approximation of solutions of nonlinear differential equations. In all cases our results are significantly better than those obtained with the maximum entropy technique for estimating \(u\) .     

Shape derivatives for minima of integral functionals

For \(\Omega \) varying among open bounded sets in \(\mathbb R ^n\) , we consider shape functionals \(J (\Omega )\) defined as the infimum over a Sobolev space of an integral energy of the kind \(\int _\Omega [ f (\nabla u) + g (u) ]\) , under Dirichlet or Neumann conditions on \(\partial \Omega \) . Under fairly weak assumptions on the integrands \(f\) and \(g\) , we prove that, when a given domain \(\Omega \) is deformed into a one-parameter family of domains \(\Omega _\varepsilon \) through an initial velocity field \(V\in W ^ {1, \infty } (\mathbb R ^n, \mathbb R ^n)\) , the corresponding shape derivative of \(J\) at \(\Omega \) in the direction of \(V\) exists. Under some further regularity assumptions, we show that the shape derivative can be represented as a boundary integral depending linearly on the normal component of \(V\) on \(\partial \Omega \) . Our approach to obtain the shape derivative is new, and it is based on the joint use of Convex Analysis and Gamma-convergence techniques. It allows to deduce, as a companion result, optimality conditions in the form of conservation laws.     

Intrinsically p-biharmonic maps

For a compact Riemannian manifold \(N\) , a domain \(\Omega \subset \mathbb {R}^m\) and for \(p\in (1, \infty )\) , we introduce an intrinsic version \(E_p\) of the \(p\) -biharmonic energy functional for maps \(u : \Omega \rightarrow N\) . This requires finding a definition for the intrinsic Hessian of maps \(u : \Omega \rightarrow N\) whose first derivatives are merely \(p\) -integrable. We prove, by means of the direct method, existence of minimizers of \(E_p\) within the corresponding intrinsic Sobolev space, and we derive a monotonicity formula. Finally, we also consider more general functionals defined in terms of polyconvex functions.     

The \bar{\partial }-Equation on an Annulus Between Two Strictly q-Convex Domains with Smooth Boundaries

In this paper, we study the global boundary regularity of the \(\bar{\partial }\) - equation on an annulus domain \(\Omega \) between two strictly \(q\) -convex domains with smooth boundaries in \(\mathbb{C }^n\) for some bidegree. To this finish, we first show that the \(\bar{\partial }\) -operator has closed range on \(L^{2}_{r, s}(\Omega )\) and the \(\bar{\partial }\) -Neumann operator exists and is compact on \(L^{2}_{r,s}(\Omega )\) for all \(r\ge 0\) , \(q\le s\le n-q- 1\) . We also prove that the \(\bar{\partial }\) -Neumann operator and the Bergman projection operator are continuous on the Sobolev space \(W^{k}_{r,s}(\Omega )\) , \(k\ge 0\) , \(r\ge 0\) , and \(q\le s\le n-q-1\) . Consequently, the \(L^{2}\) -existence theorem for the \(\bar{\partial }\) -equation on such domain is established. As an application, we obtain a global solution for the \(\bar{\partial }\) equation with Hölder and \(L^p\) -estimates on strictly \(q\) -concave domain with smooth \(\mathcal C ^2\) boundary in \(\mathbb{C }^n\) , by using the local solutions and applying the pushing out method of Kerzman (Commun Pure Appl Math 24:301–380, 1971).     

On a subsemigroup of the universal covering of Lie semigroups

Let \(G\) be a connected Lie group and \(S\) a generating Lie semigroup. An important fact is that generating Lie semigroups admit simply connected covering semigroups. Denote by \(\widetilde{S}\) the simply connected universal covering semigroup of \(S\) . In connection with the problem of identifying the semigroup \(\Gamma (S)\) of monotonic homotopy with a certain subsemigroup of the simply connected covering semigroup \(\widetilde{S}\) we consider in this paper the following subsemigroup $$\begin{aligned} \widetilde{S}_{L}=\overline{\left\langle \mathrm {Exp}(\mathbb {L} (S))\right\rangle } \subset \widetilde{S}, \end{aligned}$$ where \(\mathrm {Exp}:\mathbb {L}(S)\rightarrow S\) is the lifting to \( \widetilde{S}\) of the exponential mapping \(\exp :\mathbb {L}(S)\rightarrow S\) . We prove that \(\widetilde{S}_{L}\) is also simply connected under the assumption that the Lie semigroup \(S\) is right reversible. We further comment how this result should be related to the identification problem mentioned above.     

Optimal initial value conditions for the existence of strong solutions of the Boussinesq equations

Consider the instationary Boussinesq equations in a smooth bounded domain \(\Omega \subseteq \mathbb {R}^3\) with initial values \(u_0 \in L^2_{\sigma }(\Omega )\) , \( \theta _0 \in L^2(\Omega )\) and gravitational force \(g\) . We call \((u,\theta )\) strong solution if \((u,\theta )\) is a weak solution and additionally Serrin’s condition \(u \in L^s(0,T; L^q(\Omega ))\) holds where \( 1 satisfy \(\frac{2}{s} + \frac{3}{q} =1\) . In this paper we show that \(\int _0^{\infty } \Vert e^{-tA} u_0 \Vert _q^s \, dt < \infty \) is necessary and sufficient for the existence of such a strong solution \((u,\theta )\) in a sufficiently small interval \([0,T[\, , 0 < T\le \infty \) . Furthermore we show that strong solutions are uniquely determined and that they are smooth if the data are smooth. The crucial point is the fact that we have required no additional integrability condition for \(\theta \) in the definition of a strong solution \((u,\theta )\) .     

Pointwise convergence in Pringsheim’s sense of the summability of Fourier transforms on Wiener amalgam spaces

New multi-dimensional Wiener amalgam spaces \(W_c(L_p,\ell _\infty )(\mathbb{R }^d)\) are introduced by taking the usual one-dimensional spaces coordinatewise in each dimension. The strong Hardy-Littlewood maximal function is investigated on these spaces. The pointwise convergence in Pringsheim’s sense of the \(\theta \) -summability of multi-dimensional Fourier transforms is studied. It is proved that if the Fourier transform of \(\theta \) is in a suitable Herz space, then the \(\theta \) -means \(\sigma _T^\theta f\) converge to \(f\) a.e. for all \(f\in W_c(L_1(\log L)^{d-1},\ell _\infty )(\mathbb{R }^d)\) . Note that \(W_c(L_1(\log L)^{d-1},\ell _\infty )(\mathbb{R }^d) \supset W_c(L_r,\ell _\infty )(\mathbb{R }^d) \supset L_r(\mathbb{R }^d)\) and \(W_c(L_1(\log L)^{d-1},\ell _\infty )(\mathbb{R }^d) \supset L_1(\log L)^{d-1}(\mathbb{R }^d)\) , where \(1 . Moreover, \(\sigma _T^\theta f(x)\) converges to \(f(x)\) at each Lebesgue point of \(f\in W_c(L_1(\log L)^{d-1},\ell _\infty )(\mathbb{R }^d)\) .     

Global Calderón–Zygmund theory for nonlinear parabolic systems

We establish a global Calderón–Zygmund theory for solutions to a large class of nonlinear parabolic systems whose model is the inhomogeneous parabolic \(p\) -Laplacian system $$\begin{aligned} \left\{ \begin{array}{ll} \partial _t u - {{\mathrm{div}}}(|Du|^{p-2}Du) = {{\mathrm{div}}}(|F|^{p-2}F) &{}\quad \hbox {in }\quad \Omega _T:=\Omega \times (0,T)\\ u=g &{}\quad \hbox {on }\quad \partial \Omega \times (0,T)\cup {\overline{\Omega }}\times \{0\} \end{array} \right. \end{aligned}$$ with given functions \(F\) and \(g\) . Our main result states that the spatial gradient of the solution is as integrable as the data \(F\) and \(g\) up to the lateral boundary of \(\Omega _T\) , i.e. $$\begin{aligned} F,Dg\in L^q(\Omega _T),\ \partial _t g\in L^{\frac{q(n+2)}{p(n+2)-n}}(\Omega _T) \quad \Rightarrow \quad Du\in L^q(\Omega \times (\delta ,T)) \end{aligned}$$ for any \(q>p\) and \(\delta \in (0,T)\) , together with quantitative estimates. This result is proved in a much more general setting, i.e. for asymptotically regular parabolic systems.     

On the rank of incidence matrices in projective Hjelmslev spaces

Let \(R\) be a finite chain ring with \(|R|=q^m\) , \(R/{{\mathrm{Rad}}}R\cong \mathbb {F}_q\) , and let \(\Omega ={{\mathrm{PHG}}}({}_RR^n)\) . Let \(\tau =(\tau _1,\ldots ,\tau _n)\) be an integer sequence satisfying \(m=\tau _1\ge \tau _2\ge \cdots \ge \tau _n\ge 0\) . We consider the incidence matrix of all shape \(\varvec{m}^s=(\underbrace{m,\ldots ,m}_s)\) versus all shape \(\tau \) subspaces of \(\Omega \) with \(\varvec{m}^s\preceq \tau \preceq \varvec{m}^{n-s}\) . We prove that the rank of \(M_{\varvec{m}^s,\tau }(\Omega )\) over \(\mathbb {Q}\) is equal to the number of shape \(\varvec{m}^s\) subspaces. This is a partial analog of Kantor’s result about the rank of the incidence matrix of all \(s\) dimensional versus all \(t\) dimensional subspaces of \({{\mathrm{PG}}}(n,q)\) . We construct an example for shapes \(\sigma \) and \(\tau \) for which the rank of \(M_{\sigma ,\tau }(\Omega )\) is not maximal.     

On Sub-determinants and the Diameter of Polyhedra

We derive a new upper bound on the diameter of a polyhedron \(P = \{x {\in } {\mathbb {R}}^n :Ax\le b\}\) , where \(A \in {\mathbb {Z}}^{m\times n}\) . The bound is polynomial in \(n\) and the largest absolute value of a sub-determinant of \(A\) , denoted by \(\Delta \) . More precisely, we show that the diameter of \(P\) is bounded by \(O(\Delta ^2 n^4\log n\Delta )\) . If \(P\) is bounded, then we show that the diameter of \(P\) is at most \(O(\Delta ^2 n^{3.5}\log n\Delta )\) . For the special case in which \(A\) is a totally unimodular matrix, the bounds are \(O(n^4\log n)\) and \(O(n^{3.5}\log n)\) respectively. This improves over the previous best bound of \(O(m^{16}n^3(\log mn)^3)\) due to Dyer and Frieze (Math Program 64:1–16, 1994).     

On canonical metrics on Cartan–Hartogs domains

The Cartan–Hartogs domains are defined as a class of Hartogs type domains over irreducible bounded symmetric domains. The purpose of this paper is twofold. Firstly, for a Cartan–Hartogs domain \(\Omega ^{B^{d_0}}(\mu )\) endowed with the canonical metric \(g(\mu ),\) we obtain an explicit formula for the Bergman kernel of the weighted Hilbert space \(\mathcal {H}_{\alpha }\) of square integrable holomorphic functions on \(\left( \Omega ^{B^{d_0}}(\mu ), g(\mu )\right) \) with the weight \(\exp \{-\alpha \varphi \}\) (where \(\varphi \) is a globally defined Kähler potential for \(g(\mu )\) ) for \(\alpha >0\) , and, furthermore, we give an explicit expression of the Rawnsley’s \(\varepsilon \) -function expansion for \(\left( \Omega ^{B^{d_0}}(\mu ), g(\mu )\right) .\) Secondly, using the explicit expression of the Rawnsley’s \(\varepsilon \) -function expansion, we show that the coefficient \(a_2\) of the Rawnsley’s \(\varepsilon \) -function expansion for the Cartan–Hartogs domain \(\left( \Omega ^{B^{d_0}}(\mu ), g(\mu )\right) \) is constant on \(\Omega ^{B^{d_0}}(\mu )\) if and only if \(\left( \Omega ^{B^{d_0}}(\mu ), g(\mu )\right) \) is biholomorphically isometric to the complex hyperbolic space. So we give an affirmative answer to a conjecture raised by M. Zedda.     

Additive mappings satisfying algebraic conditions in rings

Let \(R\) be any \((n+1)!\) -torsion free ring and \(F,D: R\rightarrow R\) be additive mappings satisfying \(F(x^{n+1})=(\alpha (x))^nF(x)+\sum \nolimits _{i=1}^n (\alpha (x))^{n-i}(\beta (x))^iD(x)\) for all \(x\in R\) , where \(n\) is a fixed integer and \(\alpha \) , \(\beta \) are automorphisms of \(R\) . Then, \(D\) is Jordan left \((\alpha , \beta )\) -derivation and \(F\) is generalized Jordan left \((\alpha , \beta )\) -derivation on \(R\) and if additive mappings \(F\) and \(D\) satisfying \(F(x^{n+1})=F(x)(\alpha (x))^n+\sum \nolimits _{i=1}^n (\beta (x))^iD(x)(\alpha (x))^{n-i}\) for all \(x\in R\) . Then, \(D\) is Jordan \((\alpha , \beta )\) -derivation and \(F\) is generalized Jordan \((\alpha , \beta )\) -derivation on \(R\) . At last some immediate consequences of the above theorems have been given.     

How Far Can Chemotactic Cross-diffusion Enforce Exceeding Carrying Capacities?

We consider nonnegative solutions of the Neumann initial-boundary value problem for the chemotaxis-growth system $$\begin{aligned} \left\{ \begin{array}{l} u_t=\varepsilon u_{xx} -(uv_x)_x +ru -\mu u^2, \qquad x\in \Omega , \ t>0, \\ 0=v_{xx}-v+u, \qquad x\in \Omega , \ t>0, \end{array} \right. \quad (\star ) \end{aligned}$$ in \(\Omega :=(0,L)\subset \mathbb {R}\) with \(L>0, \varepsilon >0, r\ge 0\) and \(\mu >0\) , along with the corresponding limit problem formally obtained upon taking \(\varepsilon \searrow 0\) . For the latter hyperbolic–elliptic problem, we establish results on local existence and uniqueness within an appropriate generalized solution concept. In this context we shall moreover derive an extensibility criterion involving the norm of \(u(\cdot ,t)\) in \(L^\infty (\Omega )\) . This will enable us to conclude that in this case \(\varepsilon =0\) ,

• if \(\mu \ge 1\) , then all solutions emanating from sufficiently regular initial data are global in time, whereas
• if \(\mu <1\) , then some solutions blow-up in finite time.

The latter will reveal that the original parabolic–elliptic problem ( \(\star \) ), though known to possess no such exploding solutions, exhibits the following property of dynamical structure generation: given any \(\mu \in (0,1)\) , one can find smooth bounded initial data with the property that for each prescribed number \(M>0\) the solution of ( \(\star \) ) will attain values above \(M\) at some time, provided that \(\varepsilon \) is sufficiently small. In particular, this means that the associated carrying capacity given by \(\frac{r}{\mu }\) can be exceeded during evolution to an arbitrary extent. We finally present some numerical simulations that illustrate this type of solution behavior and that, moreover, inter alia, indicate that achieving large population densities is a transient dynamical phenomenon occurring on intermediate time scales only.     

Estimates of three classical summations on the spaces F_p~(α,q)(R~n)(0 < p ≤ 1)

We obtain the boundedness on ˙Fα,qp(Rn) for the Poisson summation and Gauss summation. Their maximal operators are proved to be bounded from˙Fα,qp(Rn) to L∞(Rn).For the maximal operator of the Bochner-Riesz summation, we prove that it is bounded from˙Fα,qp(Rn) to Lpnn-pα,∞(Rn).     

On the Dynamics of WKB Wave Functions Whose Phase are Weak KAM Solutions of H–J Equation

In the framework of toroidal Pseudodifferential operators on the flat torus \({\mathbb {T}}^n := ({\mathbb {R}} / 2\pi {\mathbb {Z}})^n\) we begin by proving the closure under composition for the class of Weyl operators \(\mathrm {Op}^w_\hbar (b)\) with symbols \(b \in S^m (\mathbb {T}^n \times \mathbb {R}^n)\) . Subsequently, we consider \(\mathrm {Op}^w_\hbar (H)\) when \(H=\frac{1}{2} |\eta |^2 + V(x)\) where \(V \in C^\infty ({\mathbb {T}}^n)\) and we exhibit the toroidal version of the equation for the Wigner transform of the solution of the Schrödinger equation. Moreover, we prove the convergence (in a weak sense) of the Wigner transform of the solution of the Schrödinger equation to the solution of the Liouville equation on \(\mathbb {T}^n \times {\mathbb {R}}^n\) written in the measure sense. These results are applied to the study of some WKB type wave functions in the Sobolev space \(H^{1} (\mathbb {T}^n; {\mathbb {C}})\) with phase functions in the class of Lipschitz continuous weak KAM solutions (positive and negative type) of the Hamilton–Jacobi equation \(\frac{1}{2} |P+ \nabla _x v (P,x)|^2 + V(x) = \bar{H}(P)\) for \(P \in \ell {\mathbb {Z}}^n\) with \(\ell >0\) , and to the study of the backward and forward time propagation of the related Wigner measures supported on the graph of \(P+ \nabla _x v\) .     

Construction of relative difference sets and Hadamard groups

‘There exist normal \((2m,2,2m,m)\) relative difference sets and thus Hadamard groups of order \(4m\) for all \(m\) of the form $$\begin{aligned} m= x2^{a+t+u+w+\delta -\epsilon +1}6^b 9^c 10^d 22^e 26^f \prod _{i=1}^s p_i^{4a_i} \prod _{i=1}^t q_i^2 \prod _{i=1}^u \left( (r_i+1)/2)r_i^{v_i}\right) \prod _{i=1}^w s_i \end{aligned}$$ under the following conditions: \(a,b,c,d,e,f,s,t,u,w\) are nonnegative integers, \(a_1,\ldots ,a_r\) and \(v_1,\ldots ,v_u\) are positive integers, \(p_1,\ldots ,p_s\) are odd primes, \(q_1,\ldots ,q_t\) and \(r_1,\ldots ,r_u\) are prime powers with \(q_i\equiv 1\ (\mathrm{mod}\ 4)\) and \(r_i\equiv 1\ (\mathrm{mod}\ 4)\) for all \(i, s_1,\ldots ,s_w\) are integers with \(1\le s_i \le 33\) or \(s_i\in \{39,43\}\) for all \(i, x\) is a positive integer such that \(2x-1\) or \(4x-1\) is a prime power. Moreover, \(\delta =1\) if \(x>1\) and \(c+s>0, \delta =0\) otherwise, \(\epsilon =1\) if \(x=1, c+s=0\) , and \(t+u+w>0, \epsilon =0\) otherwise. We also obtain some necessary conditions for the existence of \((2m,2,2m,m)\) relative difference sets in partial semidirect products of \(\mathbb{Z }_4\) with abelian groups, and provide a table cases for which \(m\le 100\) and the existence of such relative difference sets is open.     

Compactness and sequential completeness in some spaces of operators

Let \(X\) be a completely regular Hausdorff space and \(C_b(X)\) be the Banach lattice of all real-valued bounded continuous functions on \(X\) , endowed with the strict topologies \(\beta _\sigma ,\) \(\beta _\tau \) and \(\beta _t\) . Let \(\mathcal{L}_{\beta _z,\xi }(C_b(X),E)\) \((z=\sigma ,\tau ,t)\) stand for the space of all \((\beta _z,\xi )\) -continuous linear operators from \(C_b(X)\) to a locally convex Hausdorff space \((E,\xi ),\) provided with the topology \(\mathcal{T}_s\) of simple convergence. We characterize relative \(\mathcal{T}_s\) -compactness in \(\mathcal{L}_{\beta _z,\xi }(C_b(X),E)\) in terms of the representing Baire vector measures. It is shown that if \((E,\xi )\) is sequentially complete, then the spaces \((\mathcal{L}_{\beta _z,\xi }(C_b(X),E),\mathcal{T}_s)\) are sequentially complete whenever \(z=\sigma \) ; \(z=\tau \) and \(X\) is paracompact; \(z=t\) and \(X\) is paracompact and ?ech complete. Moreover, a Dieudonné–Grothendieck type theorem for operators on \(C_b(X)\) is given.     

On regularity of weak solutions to the instationary Navier–Stokes system: a review on recent results

Consider a weak instationary solution \(u\) of the Navier–Stokes equations in a domain \(\varOmega \subsetneq \mathbb {R}^3\) with Dirichlet boundary data \(u=0\) on \(\partial \Omega \) , i.e., \(u\) solves the Navier–Stokes system in the sense of distributions and $$\begin{aligned} u \in L^\infty \left( 0,T;L^2(\varOmega )\right) \cap L^2 \left( 0,T;W^{1,2}_0(\varOmega )\right) . \end{aligned}$$ Since the pioneering work of J. Leray 1933/34 it is an open problem whether weak solutions are unique and smooth. The main step—to nowadays knowledge—is to show that the given weak solution is a strong one in the sense of J. Serrin, i.e., \(u \in L^s \left( 0,T;L^q(\varOmega )\right) \) where \(s>2, q>3\) and \(\frac{2}{s}+ \frac{3}{q}=1\) . This review reports on recent progress in this important problem, considering this issue locally on an initial interval \([0,T')\) , \(T'<T\) , i.e., the problem of optimal initial values \(u(0)\) , globally on \([0,T)\) , and from a one-sided point of view \(u \in L^s \left( T'-\varepsilon ,T';L^q(\varOmega )\right) \) or \(u \in L^s\left( T',T'+\varepsilon ;L^q(\varOmega )\right) \) . Further topics deal with the energy (in-)equality, uniqueness of weak solutions, blow-up phenomena and the analysis in critical spaces for the whole space case.     

Weak lower semicontinuity for polyconvex integrals in the limit case

We prove a lower semicontinuity result for polyconvex functionals of the Calculus of Variations along sequences of maps \(u:\Omega \subset \mathbb{R }^n\rightarrow \mathbb{R }^m\) in \(W^{1,m}\) , \(2\le m\le n\) , bounded in \(W^{1,m-1}\) and convergent in \(L^1\) under mild technical conditions but without any extra coercivity assumption on the integrand.