ALBERT

Description: 〈p〉Publication date: Available online 5 July 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications〈/p〉 〈p〉Author(s): Pedro G. Massey, Noelia B. Rios, Demetrio Stojanoff〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi〉S〈/mi〉〈mo〉∈〈/mo〉〈msub〉〈mrow〉〈mi mathvariant="script"〉M〈/mi〉〈/mrow〉〈mrow〉〈mi〉d〈/mi〉〈/mrow〉〈/msub〉〈msup〉〈mrow〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="double-struck"〉C〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉+〈/mo〉〈/mrow〉〈/msup〉〈/math〉 be a positive semidefinite 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mi〉d〈/mi〉〈mo〉×〈/mo〉〈mi〉d〈/mi〉〈/math〉 complex matrix and let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈mi mathvariant="bold"〉a〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈msub〉〈mrow〉〈mo stretchy="false"〉(〈/mo〉〈msub〉〈mrow〉〈mi〉a〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈mo〉∈〈/mo〉〈msub〉〈mrow〉〈mi mathvariant="double-struck"〉I〈/mi〉〈/mrow〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈/msub〉〈/mrow〉〈/msub〉〈mo〉∈〈/mo〉〈msubsup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/mrow〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈/msubsup〉〈/math〉, indexed by 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈msub〉〈mrow〉〈mi mathvariant="double-struck"〉I〈/mi〉〈/mrow〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mo stretchy="false"〉{〈/mo〉〈mn〉1〈/mn〉〈mo〉,〈/mo〉〈mo〉…〈/mo〉〈mo〉,〈/mo〉〈mi〉k〈/mi〉〈mo stretchy="false"〉}〈/mo〉〈/math〉, be a 〈em〉k〈/em〉-tuple of positive numbers. Let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"〉〈msub〉〈mrow〉〈mi mathvariant="double-struck"〉T〈/mi〉〈/mrow〉〈mrow〉〈mi〉d〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="bold"〉a〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 denote the set of families 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"〉〈mi mathvariant="script"〉G〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈msub〉〈mrow〉〈mo stretchy="false"〉{〈/mo〉〈msub〉〈mrow〉〈mi〉g〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉}〈/mo〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈mo〉∈〈/mo〉〈msub〉〈mrow〉〈mi mathvariant="double-struck"〉I〈/mi〉〈/mrow〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈/msub〉〈/mrow〉〈/msub〉〈mo〉∈〈/mo〉〈msup〉〈mrow〉〈mo stretchy="false"〉(〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉C〈/mi〉〈/mrow〉〈mrow〉〈mi〉d〈/mi〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈/msup〉〈/math〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.svg"〉〈msup〉〈mrow〉〈mo stretchy="false"〉‖〈/mo〉〈msub〉〈mrow〉〈mi〉g〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉‖〈/mo〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈msub〉〈mrow〉〈mi〉a〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈/math〉, for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si136.svg"〉〈mi〉i〈/mi〉〈mo〉∈〈/mo〉〈msub〉〈mrow〉〈mi mathvariant="double-struck"〉I〈/mi〉〈/mrow〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈/msub〉〈/math〉; thus, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"〉〈msub〉〈mrow〉〈mi mathvariant="double-struck"〉T〈/mi〉〈/mrow〉〈mrow〉〈mi〉d〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="bold"〉a〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is the product of spheres in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉C〈/mi〉〈/mrow〉〈mrow〉〈mi〉d〈/mi〉〈/mrow〉〈/msup〉〈/math〉 endowed with the product metric. For a strictly convex unitarily invariant norm 〈em〉N〈/em〉 in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉M〈/mi〉〈/mrow〉〈mrow〉〈mi〉d〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="double-struck"〉C〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, we consider the generalized frame operator distance function 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉Θ〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉(〈/mo〉〈mi〉N〈/mi〉〈mspace width="0.2em"〉〈/mspace〉〈mo〉,〈/mo〉〈mspace width="0.2em"〉〈/mspace〉〈mi〉S〈/mi〉〈mspace width="0.2em"〉〈/mspace〉〈mo〉,〈/mo〉〈mspace width="0.2em"〉〈/mspace〉〈mi mathvariant="bold"〉a〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈/msub〉〈/math〉 defined on 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"〉〈msub〉〈mrow〉〈mi mathvariant="double-struck"〉T〈/mi〉〈/mrow〉〈mrow〉〈mi〉d〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="bold"〉a〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, given by〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.svg"〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉Θ〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉(〈/mo〉〈mi〉N〈/mi〉〈mspace width="0.2em"〉〈/mspace〉〈mo〉,〈/mo〉〈mspace width="0.2em"〉〈/mspace〉〈mi〉S〈/mi〉〈mspace width="0.2em"〉〈/mspace〉〈mo〉,〈/mo〉〈mspace width="0.2em"〉〈/mspace〉〈mi mathvariant="bold"〉a〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="script"〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo linebreak="badbreak" linebreakstyle="after"〉=〈/mo〉〈mi〉N〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉S〈/mi〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈msub〉〈mrow〉〈mi〉S〈/mi〉〈/mrow〉〈mrow〉〈mi mathvariant="script"〉G〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈mspace width="1em"〉〈/mspace〉〈mtext〉where〈/mtext〉〈mspace width="1em"〉〈/mspace〉〈msub〉〈mrow〉〈mi〉S〈/mi〉〈/mrow〉〈mrow〉〈mi mathvariant="script"〉G〈/mi〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈munder〉〈mo movablelimits="false"〉∑〈/mo〉〈mrow〉〈mi〉i〈/mi〉〈mo〉∈〈/mo〉〈msub〉〈mrow〉〈mi mathvariant="double-struck"〉I〈/mi〉〈/mrow〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈/msub〉〈/mrow〉〈/munder〉〈msub〉〈mrow〉〈mi〉g〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈mspace width="0.2em"〉〈/mspace〉〈msubsup〉〈mrow〉〈mi〉g〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈mrow〉〈mo〉⁎〈/mo〉〈/mrow〉〈/msubsup〉〈mo〉∈〈/mo〉〈msub〉〈mrow〉〈mi mathvariant="script"〉M〈/mi〉〈/mrow〉〈mrow〉〈mi〉d〈/mi〉〈/mrow〉〈/msub〉〈msup〉〈mrow〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="double-struck"〉C〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉+〈/mo〉〈/mrow〉〈/msup〉〈mspace width="0.2em"〉〈/mspace〉〈mo〉.〈/mo〉〈/math〉〈/span〉 In this paper we determine the geometrical and spectral structure of local minimizers 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉G〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mo〉∈〈/mo〉〈msub〉〈mrow〉〈mi mathvariant="double-struck"〉T〈/mi〉〈/mrow〉〈mrow〉〈mi〉d〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="bold"〉a〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉Θ〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉(〈/mo〉〈mi〉N〈/mi〉〈mspace width="0.2em"〉〈/mspace〉〈mo〉,〈/mo〉〈mspace width="0.2em"〉〈/mspace〉〈mi〉S〈/mi〉〈mspace width="0.2em"〉〈/mspace〉〈mo〉,〈/mo〉〈mspace width="0.2em"〉〈/mspace〉〈mi mathvariant="bold"〉a〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈/msub〉〈/math〉. In particular, we show that local minimizers are global minimizers, and that these families do not depend on the particular choice of 〈em〉N〈/em〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 5 July 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications〈/p〉 〈p〉Author(s): Yu-Xiang Liu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We consider a variable coefficient wave equation with an acoustic undamped condition. This is a coupled system of second and first order in time partial differential equations, with an acoustic boundary condition on the interface. The Riemannian geometry method is applied to deal with the variable coefficients. Under some checkable conditions on the coefficients we obtain the polynomial energy decay.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 5 July 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications〈/p〉 〈p〉Author(s): Barbara Kaltenbacher, Igor Shevchenko〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The focus of this work is on the analysis of the Westervelt equation modeling nonlinear propagation of high intensity ultrasound, in the practically relevant setting of a truncated computational domain with absorbing boundary conditions. We especially consider the zero and first order nonlinear absorbing boundary conditions devised in [38] in one and two space dimensions. As a matter of fact, the energy identities and estimates presented here were crucial for designing these absorbing boundary conditions in such a way that the desired energy dissipation through the boundary is guaranteed. Under the hypothesis of small initial data, we establish local well-posedness and provide higher order energy estimates, that we expect to be of additional use in boundary control and stabilization.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 5 July 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications〈/p〉 〈p〉Author(s): Alireza Ranjbar-Motlagh〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The main purpose of this article is to generalize a characterization of Lipschitz functions in the context of metric-measure spaces. The results are established in the class of metric-measure spaces which satisfy a strong version of the doubling (Bishop-Gromov regularity) condition. Indeed, we establish a necessary and sufficient condition in order that any measurable function which satisfies an integrability condition to be essentially Lipschitzian.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 5 July 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications〈/p〉 〈p〉Author(s): Rauan Akylzhanov, Michael Ruzhansky, Erlan Nursultanov〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we prove new inequalities describing the relationship between the “size” of a function on a compact homogeneous manifold and the “size” of its Fourier coefficients. These inequalities can be viewed as noncommutative versions of the Hardy-Littlewood inequalities obtained by Hardy and Littlewood [HL27] on the circle. For the example case of the group SU(2) we show that the obtained Hardy-Littlewood inequalities are sharp, yielding a criterion for a function to be in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mrow〉〈mi mathvariant="normal"〉SU〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mn mathvariant="normal"〉2〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 in terms of its Fourier coefficients. We also establish Paley and Hausdorff-Young-Paley inequalities on general compact homogeneous manifolds. The latter is applied to obtain conditions for the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈/math〉-〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉q〈/mi〉〈/mrow〉〈/msup〉〈/math〉 boundedness of Fourier multipliers for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si281.svg"〉〈mn〉1〈/mn〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mi〉p〈/mi〉〈mo〉≤〈/mo〉〈mn〉2〈/mn〉〈mo〉≤〈/mo〉〈mi〉q〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〈〈/mo〉〈mo〉∞〈/mo〉〈/math〉 on compact homogeneous manifolds as well as the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈/math〉-〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉q〈/mi〉〈/mrow〉〈/msup〉〈/math〉 boundedness of general (non-invariant) operators on compact Lie groups. We also record an abstract version of the Marcinkiewicz interpolation theorem on totally ordered discrete sets, to be used in the proofs with different Plancherel measures on the unitary duals.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 5 July 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications〈/p〉 〈p〉Author(s): Alexander Iksanov, Xingang Liang, Quansheng Liu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We prove necessary and sufficient conditions for the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈/math〉-convergence, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mi〉p〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉1〈/mn〉〈/math〉, of the Biggins martingale with complex parameter in the supercritical branching random walk. The results and their proofs are much more involved (especially in the case 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si107.svg"〉〈mi〉p〈/mi〉〈mo〉∈〈/mo〉〈mo stretchy="false"〉(〈/mo〉〈mn〉1〈/mn〉〈mo〉,〈/mo〉〈mn〉2〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈/math〉) than those for the Biggins martingale with real parameter. Our conditions are ultimate in the case 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si173.svg"〉〈mi〉p〈/mi〉〈mo〉≥〈/mo〉〈mn〉2〈/mn〉〈/math〉 only.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 5 July 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications〈/p〉 〈p〉Author(s): Sandro Zagatti〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We present an improved version of a necessary and sufficient condition for strong convergence in the Sobolev space 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi〉W〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈mo〉,〈/mo〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="normal"〉Ω〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 and provide a nontrivial application to a class of fully nonlinear partial differential equations involving the range of the map 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈msup〉〈mrow〉〈mi〉W〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈mo〉,〈/mo〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="normal"〉Ω〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉∋〈/mo〉〈mi〉u〈/mi〉〈mo stretchy="false"〉↦〈/mo〉〈mo stretchy="false"〉|〈/mo〉〈mi mathvariant="normal"〉∇〈/mi〉〈mi〉u〈/mi〉〈msup〉〈mrow〉〈mo stretchy="false"〉|〈/mo〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈mo〉∈〈/mo〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="normal"〉Ω〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 5 July 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications〈/p〉 〈p〉Author(s): Hyeonbae Kang, KiHyun Yun〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper studies in a quantitatively precise manner the field enhancement due to presence of an emitter of the dipole type near the bow-tie structure of perfectly conducting inclusions in the two-dimensional space. We put special emphasis on field enhancement near vertices of the bow-tie structure, and derive upper and lower bounds of the gradient blow-up there. All three different kinds of symmetries are considered by varying locations and directions of the emitter, and a different estimate is derived for each case.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 21 August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications〈/p〉 〈p〉Author(s): Markus Holzleitner〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The present work aims at obtaining estimates for transformation operators for one-dimensional perturbed radial Schrödinger operators. It provides more details and suitable extensions to already existing results, that are needed in other recent contributions dealing with these kinds of operators.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 21 August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Mathematical Analysis and Applications〈/p〉 〈p〉Author(s): Koichi Komada〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We study the large time asymptotics of the solutions to nonlinear dispersive equations of the form〈span〉〈span〉(0.1)〈/span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mo〉∂〈/mo〉〈/mrow〉〈mrow〉〈mi〉t〈/mi〉〈/mrow〉〈/msub〉〈mi〉u〈/mi〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mi〉P〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mi〉i〈/mi〉〈msub〉〈mrow〉〈mo〉∂〈/mo〉〈/mrow〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈msub〉〈mrow〉〈mo〉∂〈/mo〉〈/mrow〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈/msub〉〈mi〉u〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈msub〉〈mrow〉〈mo〉∂〈/mo〉〈/mrow〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉)〈/mo〉〈mo〉,〈/mo〉〈mspace width="2em"〉〈/mspace〉〈mi〉t〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mspace width="0.25em"〉〈/mspace〉〈mi〉x〈/mi〉〈mo〉∈〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈mo〉,〈/mo〉〈/math〉〈/span〉 where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mi〉P〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mi〉i〈/mi〉〈msub〉〈mrow〉〈mo〉∂〈/mo〉〈/mrow〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is a pseudo-differential operator defined by the symbol 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈mi〉P〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉ξ〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉. In this paper, we consider the case that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈mi〉P〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉ξ〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is nonlocal and nonhomogeneous and show the existence of the solution to (0.1) around a given modified free solution.〈/p〉〈/div〉