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25841 results in Abstract analysis

Page 99 of 1293

  • Inhomogeneous Helmholtz equations in wave guides – existence and uniqueness results with energy methods

  • Part of
  • BEN SCHWEIZER
  • Journal:
    European Journal of Applied Mathematics / Volume 34 / Issue 2 / April 2023
    Published online by Cambridge University Press:
    30 March 2022, pp. 211-237
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  • The Helmholtz equation $-\nabla\cdot (a\nabla u) - \omega^2 u = f$ is considered in an unbounded wave guide $\Omega := \mathbb{R} \times S \subset \mathbb{R}^d$, $S\subset \mathbb{R}^{d-1}$ a bounded domain. The coefficient a is strictly elliptic and either periodic in the unbounded direction $x_1 \in \mathbb{R}$ or periodic outside a compact subset; in the latter case, two different periodic media can be used in the two unbounded directions. For non-singular frequencies $\omega$, we show the existence of a solution u. While previous proofs of such results were based on analyticity arguments within operator theory, here, only energy methods are used.

  • THE LOCAL-ORBIFOLD CORRESPONDENCE FOR SIMPLE NORMAL CROSSING PAIRS

  • Part of
  • Luca Battistella, Navid Nabijou, Hsian-Hua Tseng, Fenglong You
  • Journal:
    Journal of the Institute of Mathematics of Jussieu / Volume 22 / Issue 5 / September 2023
    Published online by Cambridge University Press:
    25 March 2022, pp. 2515-2531
    Print publication:
    September 2023
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  • For X a smooth projective variety and $D=D_1+\dotsb +D_n$ a simple normal crossing divisor, we establish a precise cycle-level correspondence between the genus $0$ local Gromov–Witten theory of the bundle $\oplus _{i=1}^n \mathcal {O}_X(-D_i)$ and the maximal contact Gromov–Witten theory of the multiroot stack $X_{D,\vec r}$. The proof is an implementation of the rank-reduction strategy. We use this point of view to clarify the relationship between logarithmic and orbifold invariants.

Page 99 of 1293