## Recursos de colección

#### Project Euclid (Hosted at Cornell University Library) (198.174 recursos)

Differential Integral Equations

1. #### Errata to the paper “Minimization of a Ginzburg-Landau type energy with potential having a zero of infinite order”

As was kindly pointed to one of us by an anonymous referee, there is a gap in the argument in [1] whose origin is in the statement and proof of Lemma 2.2. This error can be easily corrected, and after this correction all the main results in [1] remain valid, as explained below.

2. #### Stability of the solution semigroup for neutral delay differential equations

Fabiano, Richard; Payne, Catherine
We derive a new condition for delay-independent stability of systems of linear neutral delay differential equations. The method applies ideas from linear semigroup theory, and involves renorming the underlying Hilbert space to obtain a dissipative inequality on the infinitesimal generator of the solution semigroup. The new stability condition is shown to either improve upon or be independent of existing stability conditions.

3. #### Local well-posedness of the NLS equation with third order dispersion in negative Sobolev spaces

Miyaji, Tomoyuki; Tsutsumi, Yoshio
We show the time local well-posedness in $H^s$ of the reduced NLS equation with third order dispersion (r3NLS) on $\mathbf{T}$ for $s > -1/6$. Our proof is based on the nonlinear smoothing effect, which is similar to that for mKdV. However, when (r3NLS) is considered in Sobolev spaces of negative indices, the unconditional uniqueness of solutions, that is, the uniqueness of solutions without auxiliary spaces breaks down in marked contrast to mKdV.

4. #### Removable singularities for elliptic equations with $(p,q)$-growth conditions

Namlyeyeva, Yuliya V.; Skrypnik, Igor I.
For solutions of a class of divergence type quasilinear elliptic equations with $(p,q)$-growth conditions, we establish the condition for removability of singularity on manifolds.

5. #### Symmetry breaking for an elliptic equation involving the Fractional Laplacian

Nápoli, Pablo L.
We study the symmetry breaking phenomenon for an elliptic equation involving the fractional Laplacian in a large ball. Our main tool is an extension of the Strauss radial lemma involving the fractional Laplacian, which might be of independent interest; and from which we derive compact embedding theorems for a Sobolev-type space of radial functions with power weights.

6. #### Some improvements for a class of the Caffarelli-Kohn-Nirenberg inequalities

Sano, Megumi; Takahashi, Futoshi
In this paper, we concern a weighted version of the Hardy inequality, which is a special case of the more general Caffarelli-Kohn-Nirenberg inequalities. We improve the inequality on the whole space or on a bounded domain by adding various remainder terms. On the whole space, we show the existence of a remainder term which has the form of ratio of two weighted integrals. Also we give a simple derivation of the remainder term involving a distance from the manifold of the “virtual extremals”. Finally, on a bounded domain, we prove the existence of remainder terms involving the gradient of functions.

7. #### On the lack of compactness and existence of maximizers for some Airy-Strichartz inequalities

Farah, Luiz G.; Versieux, Henrique
This work is devoted to prove a linear profile decomposition for the Airy equation in $\dot{H}_x^{s_k}(\mathbb R)$, where $s_k=(k-4)/2k$ and $k>4$. We also apply this decomposition to establish the existence of maximizers for a general class of Strichartz type inequalities associated to the Airy equation.

8. #### Steady periodic water waves bifurcating for fixed-depth rotational flows with discontinuous vorticity

David, Henry; Sastre-Gomez, Silvia
In this article, we apply local bifurcation theory to prove the existence of small-amplitude steady periodic water waves, which propagate over a flat bed with a specified fixed mean-depth, and where the underlying flow has a discontinuous vorticity distribution.

9. #### The structure of extremals of autonomous variational problems with vector-valued functions

Zaslavski, Alexander J.
In this work we study the structure of extremals of autonomous variational problems with vector-valued functions on intervals in $[0,\infty)$. We are interested in a turnpike property of the extremals which is independent of the length of the interval, for all sufficiently large intervals. To have this property means, roughly speaking, that the approximate solutions of the variational problems are determined mainly by the integrand, and are essentially independent of the choice of interval and endpoint conditions.

11. #### Quasilinear parabolic integro-differential equations with nonlinear boundary conditions

Zacher, Rico
We study the $L_p$-theory of a class of quasilinear parabolic partial integro-differential equations with nonlinear boundary conditions. The main objective here is to prove existence and uniqueness of local (in time) strong solutions of these problems. Our approach relies on linearization and the contraction mapping principle. To make this work we establish optimal regularity estimates of $L_p$ type for associated linear problems with inhomogeneous boundary data, using here recent results on maximal $L_p$-regularity for abstract parabolic Volterra equations.

12. #### Prescribing Gauss-Kronecker curvature on group invariant convex hypersurfaces

Mikula, Richard
We consider the problem of prescribing Gauss-Kronecker curvature in Euclidean space. In particular, by a degree theory argument, we prove the existence of a closed convex hypersurface in $\mathbb{R}^{3}$ which has its Gauss-Kronecker curvature equal to $F$, a prescribed positive function, which is invariant under a fixed-point free subgroup $G$ of the orthogonal group $O(3)$, requiring that $F$ satisfy natural growth assumptions near the origin and at infinity.

13. #### Non-uniform dependence on initial data for a family of non-linear evolution equations

Olson, Erika A.
We show that solutions to the periodic Cauchy problem for a family of non-linear evolution equations, which contains the Camassa-Holm equation, do not depend uniformly continuously on initial data in the Sobolev space $H^s(\mathbb{T})$, when $s=1$ or $s\geq 2$.

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