Recent zbMATH articles in MSC 35J10https://www.zbmath.org/atom/cc/35J102022-01-14T13:23:02.489162ZWerkzeugExistence of positive solution to Schrödinger-type semipositone problems with mixed nonlinear boundary conditionshttps://www.zbmath.org/1475.351282022-01-14T13:23:02.489162Z"Ko, Eunkyung"https://www.zbmath.org/authors/?q=ai:ko.eunkyung"Lee, Eun Kyoung"https://www.zbmath.org/authors/?q=ai:lee.eunkyoung"Sim, Inbo"https://www.zbmath.org/authors/?q=ai:sim.inboSummary: We studied the existence of a positive solution to Schrödinger-type semipositone problems with mixed nonlinear boundary conditions. By considering the cases when the reaction term with a parameter satisfies a superlinear and a sublinear growth condition at infinity, we established the existence of a positive solution for the large and small values of the parameter, respectively. The proofs are mainly based on the sub- and supersolution method for the sublinear case and the mountain pass lemma with \(C^{1,\alpha} (\overline{\Omega})\)-regularity for the superlinear case.Boundedness of second-order Riesz transforms on weighted Hardy and \(BMO\) spaces associated with Schrödinger operatorshttps://www.zbmath.org/1475.351292022-01-14T13:23:02.489162Z"Nguyen Ngoc, Trong"https://www.zbmath.org/authors/?q=ai:nguyen-ngoc.trong"Le Xuan, Truong"https://www.zbmath.org/authors/?q=ai:le-xuan.truong"Tan Duc, Do"https://www.zbmath.org/authors/?q=ai:tan-duc.doSummary: Let \(d\in\{3,4,5,\dots\}\) and a weight \(w\in A^\rho_\infty\). We consider the second-order Riesz transform \(T=\nabla^2\) \(L^{-1}\) associated with the Schrödinger operator \(L=-\Delta+V\), where \(V\in RH_\sigma\) with \(\sigma>\frac{d}{2}\). We present three main results. First \(T\) is bounded on the weighted Hardy space \(H^1_{w,L}(\mathbb{R}^d)\) associated with \(L\) if \(w\) enjoys a certain stable property. Secondly \(T\) is bounded on the weighted \(BMO\) space \(BMO_{w,\rho}(\mathbb{R}^d)\) associated with \(L\) if \(w\) also belongs to an appropriate doubling class. Thirdly \(BMO_{w,\rho}(\mathbb{R}^d)\) is the dual of \(H^1_{w,L}(\mathbb{R}^d)\) when \(w\in A^\rho_1\).A representation formula for the distributional normal derivativehttps://www.zbmath.org/1475.351302022-01-14T13:23:02.489162Z"Ponce, Augusto C."https://www.zbmath.org/authors/?q=ai:ponce.augusto-c"Wilmet, Nicolas"https://www.zbmath.org/authors/?q=ai:wilmet.nicolasSummary: We prove an integral representation formula for the distributional normal derivative of solutions of
\[
\begin{cases}
-\Delta u+Vu =\mu\quad&\text{in }\Omega, \\
u = 0 &\text{on }\partial\Omega,
\end{cases}
\]
where \(V\in L_{\mathrm{loc}}^1(\Omega)\) is a nonnegative function and \(\mu\) is a finite Borel measure on \(\Omega\). As an application, we show that the Hopf lemma holds almost everywhere on \(\partial\Omega\) when \(V\) is a nonnegative Hopf potential.Boundedness of operators generated by fractional semigroups associated with Schrödinger operators on Campanato type spaces via \(T1\) theoremhttps://www.zbmath.org/1475.351312022-01-14T13:23:02.489162Z"Wang, Zhiyong"https://www.zbmath.org/authors/?q=ai:wang.zhiyong.2|wang.zhiyong.1"Li, Pengtao"https://www.zbmath.org/authors/?q=ai:li.pengtao"Zhang, Chao"https://www.zbmath.org/authors/?q=ai:zhang.chao.6Summary: Let \(\mathcal{L}=-\varDelta +V\) be a Schrödinger operator, where the nonnegative potential \(V\) belongs to the reverse Hölder class \(B_q\). By the aid of the subordinative formula, we estimate the regularities of the fractional heat semigroup, \(\{e^{-t{\mathcal{L}}^{\alpha}}\}_{t>0}\), associated with \(\mathcal{L}\). As an application, we obtain the \(BMO^{\gamma}_{\mathcal{L}}\)-boundedness of the maximal function, and the Littlewood-Paley \(g\)-functions associated with \(\mathcal{L}\) via \(T1\) theorem, respectively.Existence of single peak solutions for a nonlinear Schrödinger system with coupled quadratic nonlinearityhttps://www.zbmath.org/1475.351322022-01-14T13:23:02.489162Z"Yang, Jing"https://www.zbmath.org/authors/?q=ai:yang.jing"Zhou, Ting"https://www.zbmath.org/authors/?q=ai:zhou.tingSummary: We are concerned with the following Schrödinger system with coupled quadratic nonlinearity
\[\begin{cases}
-\varepsilon^2 \Delta v+P(x) v=\mu v w, & x \in \mathbb{R}^N, \\
-\varepsilon^2 \Delta w+Q(x) w=\frac{\mu}{2} v^2+\gamma w^2, & x \in \mathbb{R}^N, \\
v>0, \quad w>0, & v, w \in H^1\left(\mathbb{R}^N\right),\end{cases}\]
which arises from second-harmonic generation in quadratic media. Here \(\epsilon > 0\) is a small parameter, \(2 \leq N < 6\), \(\mu > 0\) and \(\mu > \gamma\), \(P(x)\), \(Q(x)\) are positive function potentials. By applying reduction method, we prove that if \(x_0\) is a non-degenerate critical point of \(\Delta (P + Q)\) on some closed \(N - 1\) dimensional hypersurface, then the system above has a single peak solution \((v_\epsilon, w_\epsilon)\) concentrating at \(x_0\) for \(\epsilon\) small enough.Observability and controllability for the Schrödinger equation on quotients of groups of Heisenberg typehttps://www.zbmath.org/1475.353772022-01-14T13:23:02.489162Z"Fermanian Kammerer, Clotilde"https://www.zbmath.org/authors/?q=ai:fermanian-kammerer.clotilde"Letrouit, Cyril"https://www.zbmath.org/authors/?q=ai:letrouit.cyrilSummary: We give necessary and sufficient conditions for the controllability of a Schrödinger equation involving the sub-Laplacian of a nilmanifold obtained by taking the quotient of a group of Heisenberg type by one of its discrete sub-groups. This class of nilpotent Lie groups is a major example of stratified Lie groups of step 2. The sub-Laplacian involved in these Schrödinger equations is subelliptic, and, contrary to what happens for the usual elliptic Schrödinger equation for example on flat tori or on negatively curved manifolds, there exists a minimal time of controllability. The main tools used in the proofs are (operator-valued) semi-classical measures constructed by use of representation theory and a notion of semi-classical wave packets that we introduce here in the context of groups of Heisenberg type.A counterexample of a uniqueness resulthttps://www.zbmath.org/1475.490242022-01-14T13:23:02.489162Z"Hajaiej, H."https://www.zbmath.org/authors/?q=ai:hajaiej.hichemSummary: Uniqueness of minimizers of constrained or unconstrained energy functionals is a very subtle issue that has attracted many mathematicians in the last decades. This interest is motivated by several reasons: The uniqueness implies that the critical point inherits all the symmetry and monotonicity properties of the problem. For example if the functional and its domain are radially decreasing, then is the critical point. This considerably reduces the difficulty of the study of quantitative properties of the underlying PDE, which is reduced to an ODE. Uniqueness also ``guarantees'' the stability, and simplifies the dynamics of the gradient flow induced by the functional.
There are very few general results dealing with uniqueness of critical points in the literature. The purpose of this paper is to provide a counterexample of a general result obtained by \textit{B. Dacorogna} in his book [Introduction to the calculus of variations. Transl. from the French. River Edge, NJ: World Scientific (2004; Zbl 1095.49002)].Computing eigenvalues and eigenfunctions of Schrödinger equations using a model reduction approachhttps://www.zbmath.org/1475.651832022-01-14T13:23:02.489162Z"Li, Shuangping"https://www.zbmath.org/authors/?q=ai:li.shuangping"Zhang, Zhiwen"https://www.zbmath.org/authors/?q=ai:zhang.zhiwenSummary: We present a model reduction approach to construct problem dependent basis functions and compute eigenvalues and eigenfunctions of stationary Schrödinger equations. The basis functions are defined on coarse meshes and obtained through solving an optimization problem. We shall show that the basis functions span a low-dimensional generalized finite element space that accurately preserves the lowermost eigenvalues and eigenfunctions of the stationary Schrödinger equations. Therefore, our method avoids the application of eigenvalue solver on fine-scale discretization and offers considerable savings in solving eigenvalues and eigenfunctions of Schrödinger equations. The construction of the basis functions are independent of each other; thus our method is perfectly parallel. We also provide error estimates for the eigenvalues obtained by our new method. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method, especially Schrödinger equations with double well potentials are tested.Localization and IDS regularity in the disordered Hubbard model within Hartree-Fock theoryhttps://www.zbmath.org/1475.820112022-01-14T13:23:02.489162Z"Matos, Rodrigo"https://www.zbmath.org/authors/?q=ai:matos.rodrigo"Schenker, Jeffrey"https://www.zbmath.org/authors/?q=ai:schenker.jeffrey-hSummary: Using the fractional moment method it is shown that, within the Hartree-Fock approximation for the disordered Hubbard Hamiltonian, weakly interacting Fermions at positive temperature exhibit localization, suitably defined as exponential decay of eigenfunction correlators. Our result holds in any dimension in the regime of large disorder and at any disorder in the one dimensional case. As a consequence of our methods, we are able to show Hölder continuity of the integrated density of states with respect to energy, disorder and interaction.Stabilization for Schrödinger equation with a distributed time delay in the boundary inputhttps://www.zbmath.org/1475.930932022-01-14T13:23:02.489162Z"Cui, Haoyue"https://www.zbmath.org/authors/?q=ai:cui.haoyue"Xu, Genqi"https://www.zbmath.org/authors/?q=ai:xu.gen-qi"Chen, Yunlan"https://www.zbmath.org/authors/?q=ai:chen.yunlanSummary: In this study, the stabilization problem for Schrödinger equation with distributed input time delay is considered. The main idea of solving the stabilization problem is transformation. The original time delay system is firstly transformed into the undelayed system, and then the feedback control law which can stabilize the undelayed system is found. Finally, we prove that the feedback control law can also exponentially stabilize the time delay system.