We consider an evolution equation arising in the Peierls–Nabarro model for crystal dislocation. We study the evolution of such a dislocation function and show that, at a macroscopic scale, the ...dislocations have the tendency to concentrate at single points of the crystal, where the size of the slip coincides with the natural periodicity of the medium. These dislocation points evolve according to the external stress and an interior repulsive potential.
We extend the classical Bernstein technique to the setting of integro-differential operators. As a consequence, we provide first and one-sided second derivative estimates for solutions to fractional ...equations, including some convex fully nonlinear equations of order smaller than two—for which we prove uniform estimates as their order approaches two. Our method is robust enough to be applied to some Pucci-type extremal equations and to obstacle problems for fractional operators, although several of the results are new even in the linear case. We also raise some intriguing open questions, one of them concerning the “pure” linear fractional Laplacian, another one being the validity of one-sided second derivative estimates for Pucci-type convex equations associated to linear operators with general kernels.
We consider the first Dirichlet eigenvalue problem for a mixed local/nonlocal elliptic operator and we establish a quantitative Faber-Krahn inequality. More precisely, we show that balls minimize the ...first eigenvalue among sets of given volume and we provide a stability result for sets that almost attain the minimum.
We prove a simple sufficient criterion to obtain some Hardy inequalities on Riemannian manifolds related to quasilinear second order differential operator Δpu:=div(|∇u|p−2∇u). Namely, if ρ is a ...nonnegative weight such that −Δpρ⩾0, then the Hardy inequalityc∫M|u|pρp|∇ρ|pdvg⩽∫M|∇u|pdvg,u∈C0∞(M), holds. We show concrete examples specializing the function ρ.
Our approach allows to obtain a characterization of p-hyperbolic manifolds as well as other inequalities related to Caccioppoli inequalities, weighted Gagliardo–Nirenberg inequalities, uncertain principle and first order Caffarelli–Kohn–Nirenberg interpolation inequality.
For
1
<
p
<
∞
, we prove radial symmetry for bounded nonnegative solutions of
-
div
w
(
x
)
H
(
∇
u
)
p
-
1
∇
ξ
H
(
∇
u
)
=
f
(
u
)
w
(
x
)
in
Σ
∩
Ω
,
u
=
0
on
Γ
0
,
⟨
∇
ξ
H
(
∇
u
)
,
ν
⟩
=
0
on
Γ
1
...\
0
,
where
Ω
is a Wulff ball,
Σ
is a convex cone with vertex at the center of
Ω
,
Γ
0
:
=
Σ
∩
∂
Ω
,
Γ
1
:
=
∂
Σ
∩
Ω
,
H
is a norm,
w
is a given weight and
f
is a possibly discontinuous nonnegative nonlinearity.
Given the anisotropic setting that we deal with, the term “radial” is understood in the Finsler framework, that is, the function
u
is radial if there exists a point
x
such that
u
is constant on the Wulff shapes centered at
x
.
When
Σ
=
R
N
, J. Serra obtained the symmetry result in the isotropic unweighted setting (i.e., when
H
(
ξ
)
≡
|
ξ
|
and
w
≡
1
). In this case we provide the extension of his result to the anisotropic setting. This provides a generalization to the anisotropic setting of a celebrated result due to Gidas-Ni-Nirenberg and such a generalization is new even for
p
=
2
whenever
N
>
2
. When
Σ
⊊
R
N
the results presented are new even in the isotropic and unweighted setting (i.e., when
H
is the Euclidean norm and
w
≡
1
) whenever
2
≠
p
≠
N
. Even for the previously known case of unweighted isotropic setting with
p
=
2
and
Σ
⊊
R
N
, the present paper provides an approach to the problem by exploiting integral (in)equalities which is new for
N
>
2
: this complements the corresponding symmetry result obtained via the moving planes method by Berestycki-Pacella.
The results obtained in the isotropic and weighted setting (i.e., with
w
≢
1
) are new for any
p
.
We prove that nonlocal minimal graphs in the plane exhibit generically stickiness effects and boundary discontinuities. More precisely, we show that if a nonlocal minimal graph in a slab is ...continuous up to the boundary, then arbitrarily small perturbations of the far-away data produce boundary discontinuities. Hence, either a nonlocal minimal graph is discontinuous at the boundary, or a small perturbation of the prescribed conditions produces boundary discontinuities. The proof relies on a sliding method combined with a fine boundary regularity analysis, based on a discontinuity/smoothness alternative. Namely, we establish that nonlocal minimal graphs are either discontinuous at the boundary or their derivative is Hölder continuous up to the boundary. In this spirit, we prove that the boundary regularity of nonlocal minimal graphs in the plane “jumps” from discontinuous to
C
1
,
γ
, with no intermediate possibilities allowed. In particular, we deduce that the nonlocal curvature equation is always satisfied up to the boundary. As an interesting byproduct of our analysis, one obtains a detailed understanding of the “switch” between the regime of continuous (and hence differentiable) nonlocal minimal graphs to that of discontinuous (and hence with differentiable inverse) ones.
Napoleonic Triangles on the Sphere Dipierro, Serena; Noakes, Lyle; Valdinoci, Enrico
Boletim da Sociedade Brasileira de Matemática,
06/2024, Letnik:
55, Številka:
2
Journal Article
Recenzirano
Odprti dostop
As is well-known, numerical experiments show that Napoleon’s Theorem for planar triangles does not extend to a similar statement for triangles on the unit sphere
S
2
. Spherical triangles for which ...an extension of Napoleon’s Theorem holds are called
Napoleonic
, and until now the only known examples have been equilateral. In this paper we determine all Napoleonic spherical triangles, including a class corresponding to points on a 2-dimensional ellipsoid, whose Napoleonisations are all congruent. Other new classes of examples are also found, according to different versions of Napoleon’s Theorem for the sphere. The classification follows from successive simplifications of a complicated original algebraic condition, exploiting geometric symmetries and algebraic factorisations.
We study symmetry and quantitative approximate symmetry for an overdetermined problem involving the fractional torsion problem in a bounded open set \Omega \subset \mathbb R^n. More precisely, we ...prove that if the fractional torsion function has a C^1 level surface which is parallel to the boundary \partial \Omega then \Omega is a ball. If instead we assume that the solution is close to a constant on a parallel surface to the boundary, then we quantitatively prove that \Omega is close to a ball. Our results use techniques which are peculiar of the nonlocal case as, for instance, quantitative versions of fractional Hopf boundary point lemma and boundary Harnack estimates for antisymmetric functions. We also provide an application to the study of rural-urban fringes in population settlements.
We study a nonlocal capillarity problem with interaction kernels that are possibly anisotropic and not necessarily invariant under scaling. In particular, the lack of scale invariance will be modeled ...via two different fractional exponents
s
1
,
s
2
∈
(
0
,
1
)
which take into account the possibility that the container and the environment present different features with respect to particle interactions. We determine a nonlocal Young’s law for the contact angle and discuss the unique solvability of the corresponding equation in terms of the interaction kernels and of the relative adhesion coefficient.
We consider nonlocal minimal surfaces in a cylinder with prescribed datum given by the complement of a slab. We show that when the width of the slab is large the minimizers are disconnected and when ...the width of the slab is small the minimizers are connected. This feature is in agreement with the classical case of the minimal surfaces.
Nevertheless, we show that when the width of the slab is large the minimizers are not flat discs, as it happens in the classical setting, and, in particular, in dimension 2 we provide a quantitative bound on the stickiness property exhibited by the minimizers.
Moreover, differently from the classical case, we show that when the width of the slab is small then the minimizers completely adhere to the side of the cylinder, thus providing a further example of stickiness phenomenon.
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