1,720,974 research outputs found
Periodic minimal surfaces embedded in R^3 derived from the singly periodic Scherk minimal surface
No full textWe construct three kinds of periodic minimal surfaces embedded in R^3. We show the existence of a 1-parameter family of minimal surfaces invariant under the action of a translation by 2π, which seen from a distance look like m equidistant parallel planes intersecting orthogonally k equidistant parallel planes, m,k ε N, mk ≥ 2. We also consider the case where the surfaces are asymptotic to m ε+ equidistant parallel planes intersecting orthogonally infinitely many equidistant parallel planes. In this case, the minimal surfaces are doubly periodic, precisely they are invariant under the action of two orthogonal translations. Last we construct triply periodic minimal surfaces which are invariant under the action of three orthogonal translations in the case of two stacks of infinitely many equidistant parallel planes which intersect orthogonally
Symmetry breaking bifurcations for two overdetermined boundary value problems with non-constant Neumann condition on exterior domains in R3
No full textWe study two overdetermined elliptic boundary value problems on exterior domains (the complement of a ball and the complement of a solid cylinder in (Formula presented.) respectively). The Neumann condition is non-constant and involves the mean curvature of the boundary. We show there exists a family of bifurcation branches of domains which are small deformations of the complement of a ball and of the complement of a solid cylinder, respectively, and which support the solution of the overdetermined boundary value problem
Free boundaries surfaces and Saddle towers minimal surfaces in S^2 x R
No full textThe aim of this work is to show that for each finite natural number l⩾2 there exists a 1-parameter family of Saddle Tower type minimal surfaces embedded in S^2×R, invariant with respect to a vertical translation. The genus of the quotient surface is 2l−1. The proof is based on analytical techniques: precisely we desingularize of the union of γ_j×R, j∈{1,...,2l}, where γ_j⊂S^2 denotes a half great circle. These vertical cylinders intersect along a vertical straight line and its antipodal line. As byproduct of the construction we produce free boundary surfaces embedded in (S^2)^+×R. Such surfaces are extended by reflection in ∂(S^2)+×R in order to get the minimal surfaces with the desired properties
Bounded and unbounded capillary surfaces derived from the catenoid
No full textWe construct two kinds of capillary surfaces by using a perturbation method. Surfaces of first kind are embedded in a solid ball B of R3 with assigned mean curvature function and whose boundary curves lie on the boundary of B: The contact angle along such curves is a non-constant function. Surfaces of second kind are unbounded and embedded in R3 \ B'; B' being a deformation of a solid ball in R3: These surfaces have assigned mean curvature function and one boundary curve on the boundary of B' : Also in this case the contact angle along the boundary is a non-constant function
Radial and non-radial solutions to an elliptic problem on annular domains in Riemannian manifolds with radial symmetry
No full textWe show existence and uniqueness of positive radial solutions to. {Delta_g u+λu+u^p=0 in A,u=0 on ∂A, with λ. < 0, A being an annular domain in a Riemannian manifold M of dimension n endowed with the metric dr^2+S^2(r)g_(S^n-1). Secondly we show that there exist positive non-radial solutions arising by bifurcation from the radial solution. p and λ are the bifurcation parameters
Singly periodic free boundary minimal surfaces in a solid cylinder of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml2" display="inline" overflow="scroll" altimg="si2.gif"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>×</mml:mo><mml:mi mathvariant="double-struck">R</mml:mi></mml:math>
No full textThe aim of this work is to show there exist free boundary minimal surfaces of Saddle Tower type which are embedded in a vertical solid cylinder of H^2×R, H^2 being the hyperbolic plane, and invariant under the action of a vertical translation and a rotation. The number of boundary curves equals 2l,l⩾2. These surfaces come in families depending on one parameter and they converge to 2l vertical stripes having a common intersection line
Asymptotically radial solutions to an elliptic problem on expanding annular domains in Riemannian manifolds with radial symmetry
Get PDFWe consider the boundary value problem (Formula presented.) Ω_R being a smooth bounded domain diffeomorphic to the expanding domain A_R: = { x∈ M, R< r(x) < R+ 1 } in a Riemannian manifold M of dimension n≥ 2 endowed with the metric g=dr^2+S^2(r)g_(S^n−1). After recalling a result about existence, uniqueness, and non-degeneracy of the positive radial solution when Ω_R= A_R, we prove that there exists a positive non-radial solution to the aforementioned problem on the domain Ω_R. Such a solution is close to the radial solution to the corresponding problem on A_R
Height estimate for special Weingarten surfaces of elliptic type in
No full textIn this article we provide a vertical height
estimate for compact special Weingarten surfaces of elliptic
type in ²()×ℝ, i.e. surfaces whose mean curvature ℍ and
extrinsic Gauss curvature ₑ satisfy ℍ=(ℍ²-ₑ) with
4(’())²<1, for all ∈[0,+∞). The vertical height estimate
generalizes a result by Rosenberg and Sa Earp and applies only
to surfaces verifying a height estimate condition. When <0,
using also a horizontal height estimate, we show a non-existence
result for properly embedded Weingarten surfaces of elliptic
type in ℍ²()×ℝ with finite topology and one end
Symmetry breaking bifurcations for an overdetermined boundary value problem on an exterior domain issued from electrodynamics
No full textWe consider an electrically charged fluid occupying a solid cylindrical region Ω of R^3. Outside the domain Ω there is an electric field with electric potential which solves the Laplace equation and diverges as the distance from the axis tends to infinity. At ∂Ω the potential is constant and there is a balance between the pressure difference inside and outside the fluid, capillary forces proportional to the mean curvature and electrostatic repulsion of charges. We are interested in showing the existence of domains different from the solid cylinder Ω and satisfying the conditions described above. This problem is equivalent to an overdetermined elliptic boundary value problem on an exterior domain. We show the bifurcation phenomenon occurs and produces the deformation of the solid cylinder into rippled cylinders
- …
