Posibile directii de cercetare si cateva probleme deschise care au aparut pe parcursul obtinerii rezul- tatelor prezentate in aceasta lucrare au fost… [619782]

Chapter 4
Further Developments
Posibile directii de cercetare si cateva probleme deschise care au aparut pe parcursul obtinerii rezul-
tatelor prezentate in aceasta lucrare au fost deja mentionate in cadrul acesteia. Altele le voi mentionain cele ce urmeaza.
•Problem 1. In Section 1.3 we studied some 1 −dimensional models for convection–diffusion
processes in which the characters of both diffusion and convection change discontinuouslyat an internal domain point, there is a small parameter ε, making them singular perturbation
problems, and the boundary conditions are linear (Subsections 1.3.1 and 1.3.2) or one of themis nonlinear (Subsection 1.3.2). The stationary and evolutionary, 2 or n−dimensional cases
of these models could be investigated using Vishik-Lyusternik method. These results willprovide the broad theoretical base from which useful numerical and experimental engineeringmethods and new information may be gained.
•Problem 2. Regarding the elliptic regularizations investigated in Section 2.1, for the case
of linear maximal operator A, we can try to obtain similar results for the nonlinear case of
operator A.. These results could open the way for applications to many problems, including
initial-boundary value problems like that obtained in Sections 2.3, 2.4 and much more forthose associated with semilinear hyperbolic equations or systems with nonlinear boundary
conditions.
•Problem 3. In Sections 2.1 and 2.2, two elliptic regularizations for a Cauchy problem in
a Hibert space have been investigated. We can try to obtain similar results for hyperbolicregularizations of the same problem (dupa modelul celor din [15, Part IV]).
•Problem 4. In L. Barbu-G. Moros ¸anu-C. Pintea [22] has been studied an eigenvalue–transmission
problem, more exactly:
⎧
⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎩−Δ
pu1=λ|u1|p−2u1inΩ1, (1)
−Δqu2=λ|u2|q−2u2inΩ2, (2)
∂u1
∂ν p=0o nΓ1,∂u2
∂ν q=0o nΓ2, (3)
u1=u2,∂u1
∂ν p=∂u2
∂ν qonΓ, (4)
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Bibliography
where Ω⊂RN,N≥2 is a bounded domain with Lipschitz boundary ∂Ω, which is divided into
two Lipschitz sub-domains, Ω1andΩ2, by a Lipschitz closed hypersurface H,Ω=Ω1∪Ω2∪
Γ, withΓ=H∩Ω,∂Ω1=Γ1∪Γ, and ∂Ω2=Γ2∪Γ. Using the Lusternik–Schnirelmann prin-
ciple we proved the existence of a sequence of eigenvalues for the above eigenvalue problem.Our objectives here are to obtain: 1)similar results for mixed boundary conditions (Dirichlet-
Neumann or Neumann-Robin etc.); 2)a similar investigation will be done when we deal with
more general fully nonlinear operators, for example with the p-Finsler Laplacian acting in Ω
1
and the q-Finsler Laplacian acting in Ω2.
•Problem 5. In L. Barbu-G. Moros ¸anu [21] an eigenvalue problem for the negative (p,q)−
Laplacian with a Steklov-type boundary condition, where 1 <p<∞,2<q<∞,p<q,
is studied. A full description of the set of eigenvalues of this problem was provided. Ourobjectives here are to: 1)show that such results should be true for all 1 <p<q;2)see if the
same results remain true in the case of the negative (p,q)Finsler Laplacian; 3)extend certain
results from S. Barile, G. M. Figueiredo [28] for a class of general quasilinear anisotropicequations, of type celor din Section 3.2.
•Problem 6. In his pioneering paper [76], L. Modica proved the following Liouville-type
theorem: ”If F∈C
2(R)is a non-negative function, f=F/prime,uis a C3(RN)bounded solution
in all of RNof the Poisson equation /triangleu= f(u)and there exists a point x0∈RNsuch that
F(u(x0)) = 0,then u is constant.” This result was later extended to some nonlinear pde’s in
divergence form by L. Caffarelli-N. Garofalo-F. Segala [37]. In [37], the authors also provedthe following DeGiorgi type result: ”if the P-function involved in their proofs has a zero, thenthere exist a real function g,a real number aand vector bsuch that u(x)= g(a+bx).Our
aims are to: 1)show that a Modica type result is also true for some fully nonlinear pde’s (at
least for Monge-Ampere equations), with a slight natural modification of the condition on F;2)show that the above DeGiorgi type result might also be true for k-hessian equations; 3)see
if a DeGiorgi type result might be obtained for parabolic pde’s.
•Problem 7. C. Bianchini and P . Salani proved in [33] some concavity results for nonlinear
Bernoulli type free boundary problems. First, they proved a Brunn-Minkowski inequality forthe Bernoulli constant. Then, as a Corollary, they obtained the isoperimetric inequality: amongall (convex) sets with fixed mean width, the Bernoulli constant attains its minimum on balls.Finally, a concavity behaviour of the free boundary (both for the exterior and interior case)have been shown. Our aim here is to investigate if similar geometric results remains true whenwe deal with anisotropic Bernoulli type problems.
•Problem 8. A. Henrot and G.A. Philippin studied in [62] three classes of overdetermined
boundary value problems in the plane: a problem of torsion, a problem of electrostatic capacityand a problem of polarization. In each case they proved that a solution exists if and only if thefree boundary is an ellipse. Moreover, in the same paper they conjectured that extensions oftheir results to R
3should also be true. Our main aim here is to prove the conjectures proposed
in [62] in any dimension.
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