We study existence, regularity, and qualitative properties of solutions to the system−Δu=|v|q−1v in Ω,−Δv=|u|p−1u in Ω,∂νu=∂νv=0 on ∂Ω, with Ω⊂RN bounded; in this setting, all nontrivial solutions ...are sign changing. Our proofs use a variational formulation in dual spaces, considering sublinear pq<1 and superlinear pq>1 problems in the subcritical regime. In balls and annuli we show that least energy solutions (l.e.s.) are foliated Schwarz symmetric and, due to a symmetry-breaking phenomenon, l.e.s. are not radial functions; a key element in the proof is a new Lt-norm-preserving transformation, which combines a suitable flipping with a decreasing rearrangement. This combination allows us to treat annular domains, sign-changing functions, and Neumann problems, which are non-standard settings to use rearrangements and symmetrizations. In particular, we show that our transformation diminishes the (dual) energy and, as a consequence, radial l.e.s. are strictly monotone. We also study unique continuation properties and simplicity of zeros. Our theorems also apply to the scalar associated model, where our approach provides new results as well as alternative proofs of known facts.
We consider the equation −Δu=|x|α|u|p−1u for any α≥0, either in R2 or in the unit ball B of R2 centered at the origin with Dirichlet or Neumann boundary conditions. We give a sharp description of the ...asymptotic behavior as p→+∞ of all the radial solutions to these problems and we show that there is no uniform a priori bound for nodal solutions under Neumann or Dirichlet boundary conditions. This contrasts with the existence of uniform bounds for positive solutions, as shown in 32 for α=0 and Dirichlet boundary conditions.
We consider positive solutions of a fractional Lane–Emden-type problem in a bounded domain with Dirichlet conditions. We show that uniqueness and nondegeneracy hold for the asymptotically linear ...problem in general domains. Furthermore, we also prove that all the known uniqueness and nondegeneracy results in the local case extend to the nonlocal regime when the fractional parameter s is sufficiently close to 1.
We analyze the s-dependence of solutions us to the family of fractional Poisson problems(−Δ)su=fin Ω,u≡0on RN∖Ω in an open bounded set Ω⊂RN, s∈(0,1). In the case where Ω is of class C2 and f∈Cα(Ω‾) ...for some α>0, we show that the map (0,1)→L∞(Ω), s↦us is of class C1, and we characterize the derivative ∂sus in terms of the logarithmic Laplacian of f. As a corollary, we derive pointwise monotonicity properties of the solution map s↦us under suitable assumptions on f and Ω. Moreover, we derive explicit bounds for the corresponding Green operator on arbitrary bounded domains which are new even for the case s=1, i.e., for the local Dirichlet problem −Δu=f in Ω, u≡0 on ∂Ω.
We study the symmetry properties of limit profiles of nonautonomous nonlinear parabolic systems with Dirichlet boundary conditions in radial bounded domains. In the case of competitive systems, we ...show that if the initial profiles satisfy a reflectional inequality with respect to a hyperplane, then all limit profiles are foliated Schwarz symmetric with respect to antipodal points. One of the main ingredients in the proof is a new parabolic version of Serrin’s boundary point lemma. Results on radial symmetry of semi-trivial profiles are discussed also for noncompetitive systems.
The ball-burnishing process involves a surface plastic deformation that improves the physical-mechanical properties of manufactured parts. In this work, was used to study the influence of the main ...parameters of the ball-burnishing process on features such as mean surface roughness and hardness of cylindrical AISI 1045 steel samples. The experimental stage was based on a 33 factorial design and the Response Surface Methodology. An equation was proposed to model the mean roughness in the studied experimental region. Moreover, the polarization technique (Tafel) and electrochemical impedance spectroscopy (EIS) were used to evaluate the burnishing process effect on corrosion resistance of a sample machined with the optimal burnishing parameters obtained herein. Finally, burnishing process effect on the phase change was also evaluated by using X-ray diffraction. As a result, the factor that showed higher influence was the burnishing force for both features, roughness and hardness. The data shows that it is possible to reduce the surface roughness from 3.51 μm to 0.61 μm and to increase the hardness from 202 HB to 236 HB using ball-burnishing process, and to improve the corrosion resistance in AISI-1045 steel.
•Surface mean roughness enhanced 83% on optimal burnished AISI 1045 steel sample.•Hardness on AISI 1045 steel samples improved 14% with optimal burnishing parameters.•A prediction model is presented for the surface mean roughness with an error of 8%.•Tafel and EIS indicate an increase on corrosion resistance in burnished samples.•X-ray diffraction suggests an increase of d-spacing due to burnishing process.
We study the pure Neumann Lane–Emden problem in a bounded domain
-
Δ
u
=
|
u
|
p
-
1
u
in
Ω
,
∂
ν
u
=
0
on
∂
Ω
,
in the subcritical, critical, and supercritical regimes. We show existence and ...convergence of least-energy (nodal) solutions (l.e.n.s.). In particular, we prove that l.e.n.s. converge to a l.e.n.s. of a problem with sign nonlinearity as
p
↘
0
; to a l.e.n.s. of the critical problem as
p
↗
2
∗
-
1
(in particular, pure Neumann problems exhibit no blowup phenomena at the critical Sobolev exponent); and we show that the limit as
p
→
1
depends on the domain. Our proofs rely on different variational characterizations of solutions including a dual approach and a nonlinear eigenvalue problem. Finally, we also provide a qualitative analysis of l.e.n.s., including symmetry, symmetry-breaking, and monotonicity results for radial solutions.
We study the asymptotic behavior of solutions to various Dirichlet sublinear-type problems involving the fractional Laplacian when the fractional parameter
s
tends to zero. Depending on the type on ...nonlinearity, positive solutions may converge to a characteristic function or to a positive solution of a limit nonlinear problem in terms of the logarithmic Laplacian, that is, the pseudodifferential operator with Fourier symbol
ln
(
|
ξ
|
2
)
. In the case of a logistic-type nonlinearity, our results have the following biological interpretation: in the presence of a toxic boundary, species with reduced mobility have a lower saturation threshold, higher survival rate, and are more homogeneously distributed. As a result of independent interest, we show that sublinear logarithmic problems have a unique least-energy solution, which is bounded and Dini continuous with a log-Hölder modulus of continuity.
Abstract
We study an optimal $M$-partition problem for the Yamabe equation on the round sphere, in the presence of some particular symmetries. We show that there is a correspondence between solutions ...to this problem and least energy sign-changing symmetric solutions to the Yamabe equation on the sphere with precisely $M$ nodal domains.
The existence of an optimal partition is established through the study of the limit profiles of least energy solutions to a weakly coupled competitive elliptic system on the sphere.
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