| 1. |
- Ackermann, Nils, et al.
(författare)
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A concentration phenomenon for semilinear elliptic equations
- 2013
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Ingår i: Archive for Rational Mechanics and Analysis. - : Springer Science and Business Media LLC. - 0003-9527 .- 1432-0673. ; 207:3, s. 1075-1089
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Tidskriftsartikel (refereegranskat)abstract
- For a domain $\Omega\subset\dR^N$ we consider the equation $ -\Delta u + V(x)u = Q_n(x)\abs{u}^{p-2}u$ with zero Dirichlet boundary conditions and $p\in(2,2^*)$. Here $V\ge 0$ and $Q_n$ are bounded functions that are positive in a region contained in $\Omega$ and negative outside, and such that the sets $\{Q_n>0\}$ shrink to a point $x_0\in\Omega$ as $n\to\infty$. We show that if $u_n$ is a nontrivial solution corresponding to $Q_n$, then the sequence $(u_n)$ concentrates at $x_0$ with respect to the $H^1$ and certain $L^q$-norms. We also show that if the sets $\{Q_n>0\}$ shrink to two points and $u_n$ are ground state solutions, then they concentrate at one of these points.
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| 2. |
- Aleksanyan, Hayk, et al.
(författare)
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Applications of Fourier Analysis in Homogenization of the Dirichlet Problem : L-p Estimates
- 2015
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Ingår i: Archive for Rational Mechanics and Analysis. - : Springer. - 0003-9527 .- 1432-0673. ; 215:1, s. 65-87
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Tidskriftsartikel (refereegranskat)abstract
- Let u(epsilon) be a solution to the system div(A(epsilon)(x)del u(epsilon)(x)) = 0 in D, u(epsilon)(x) = g(x, x/epsilon) on partial derivative D, where D subset of R-d (d >= 2), is a smooth uniformly convex domain, and g is 1-periodic in its second variable, and both A(epsilon) and g are sufficiently smooth. Our results in this paper are twofold. First we prove L-p convergence results for solutions of the above system and for the non-oscillating operator A(epsilon)(x) = A(x), with the following convergence rate for all 1 <= p < infinity parallel to u(epsilon) - u(0)parallel to (LP(D)) <= C-P {epsilon(1/2p), d = 2, (epsilon vertical bar ln epsilon vertical bar)(1/p), d = 3, epsilon(1/p), d >= 4, which we prove is (generically) sharp for d >= 4. Here u(0) is the solution to the averaging problem. Second, combining our method with the recent results due to Kenig, Lin and Shen (Commun Pure Appl Math 67(8): 1219-1262, 2014), we prove (for certain class of operators and when d >= 3) ||u(epsilon) - u(0)||(Lp(D)) <= C-p[epsilon(ln(1/epsilon))(2)](1/p) for both the oscillating operator and boundary data. For this case, we take A(epsilon) = A(x/epsilon), where A is 1-periodic as well. Some further applications of the method to the homogenization of the Neumann problem with oscillating boundary data are also considered.
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| 3. |
- Aleman, Alexandru, et al.
(författare)
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Harmonic Maps and Ideal Fluid Flows
- 2012
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Ingår i: Archive for Rational Mechanics and Analysis. - : Springer Science and Business Media LLC. - 0003-9527 .- 1432-0673. ; 204:2, s. 479-513
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Tidskriftsartikel (refereegranskat)abstract
- Using harmonic maps we provide an approach towards obtaining explicit solutions to the incompressible two-dimensional Euler equations. More precisely, the problem of finding all solutions which in Lagrangian variables (describing the particle paths of the flow) present a labelling by harmonic functions is reduced to solving an explicit nonlinear differential system in C-n with n = 3 or n = 4. While the general solution is not available in explicit form, structural properties of the system permit us to identify several families of explicit solutions.
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| 4. |
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| 5. |
- Alho, A., et al.
(författare)
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Static Self-gravitating Newtonian Elastic Balls
- 2020
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Ingår i: Archive for Rational Mechanics and Analysis. - : Springer Science and Business Media LLC. - 0003-9527 .- 1432-0673. ; 238, s. 639-669
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Tidskriftsartikel (refereegranskat)abstract
- The existence of static self-gravitating Newtonian elastic balls is proved under general assumptions on the constitutive equations of the elastic material. The proof uses methods from the theory of finite-dimensional dynamical systems and the Euler formulation of elasticity theory for spherically symmetric bodies introduced recently by the authors. Examples of elastic materials covered by the results of this paper are Saint Venant-Kirchhoff, John and Hadamard materials.
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| 6. |
- Alkhutov, Yurij A., et al.
(författare)
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Boyarsky-Meyers Estimate for Solutions to Zaremba Problem
- 2022
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Ingår i: Archive for Rational Mechanics and Analysis. - : SPRINGER. - 0003-9527 .- 1432-0673. ; 245:2, s. 1197-1211
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Tidskriftsartikel (refereegranskat)abstract
- The variational solution to the Zaremba problem for a divergent linear second order elliptic equation with measurable coefficients is considered. The problem is set in a local Lipschitz graph domain. An estimate in L2+δ, δ > 0, for the gradient of a solution, is proved. An example of the problem with the Dirichlet data supported by a fractal set of zero (n - 1)-dimensional measure and non-zero p-capacity, p > 1 is constructed.
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| 7. |
- Allen, Mark, et al.
(författare)
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A New Boundary Harnack Principle (Equations with Right Hand Side)
- 2019
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Ingår i: Archive for Rational Mechanics and Analysis. - : SPRINGER. - 0003-9527 .- 1432-0673. ; 234:3, s. 1413-1444
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Tidskriftsartikel (refereegranskat)abstract
- We introduce a new boundary Harnack principle in Lipschitz domains for equations with a right hand side. Our approach, which uses comparisons and blow-ups, will adapt to more general domains as well as other types of operators. We prove the principle for divergence form elliptic equations with lower order terms including zero order terms. The inclusion of a zero order term appears to be new even in the absence of a right hand side.
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| 8. |
- Andersson, John, et al.
(författare)
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Double Obstacle Problems with Obstacles Given by Non-C-2 Hamilton-Jacobi Equations
- 2012
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Ingår i: Archive for Rational Mechanics and Analysis. - : Springer Science and Business Media LLC. - 0003-9527 .- 1432-0673. ; 206:3, s. 779-819
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Tidskriftsartikel (refereegranskat)abstract
- We prove optimal regularity for double obstacle problems when obstacles are given by solutions to Hamilton-Jacobi equations that are not C (2). When the Hamilton-Jacobi equation is not C (2) then the standard Bernstein technique fails and we lose the usual semi-concavity estimates. Using a non-homogeneous scaling (different speeds in different directions) we develop a new pointwise regularity theory for Hamilton-Jacobi equations at points where the solution touches the obstacle. A consequence of our result is that C (1)-solutions to the Hamilton-Jacobi equation , are, in fact, C (1,alpha/2), provided that . This result is optimal and, to the authors' best knowledge, new.
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| 9. |
- Andreasson, Håkan, 1966, et al.
(författare)
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Comments on the paper ‘Static solutions of the Vlasov–Einstein system’ by G. Wolansky
- 2020
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Ingår i: Archive for Rational Mechanics and Analysis. - : Springer Science and Business Media LLC. - 1432-0673 .- 0003-9527. ; 235:1, s. 783-791
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Tidskriftsartikel (refereegranskat)abstract
- In this note we address the attempted proof of the existence of static solutions to the Einstein–Vlasov system as given in Wolansky (Arch Ration Mech Anal 156:205–230, 2001). We focus on a specific and central part of the proof which concerns a variational problem with an obstacle. We show that two important claims in Wolansky (2001) are incorrect and we question the validity of a third claim. We also discuss the variational problem and its difficulties with the aim of stimulating further investigation of this intriguing problem in particular answering the question of whether or not static solutions of the Einstein–Vlasov system can be found as local minimizers of an energy-Casimir functional.
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| 10. |
- Arkeryd, Leif, 1940, et al.
(författare)
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Stability for Rayleigh-Benard convective solutions of the Boltzmann equation
- 2010
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Ingår i: Archive for rational mechanics and analysis. - : Springer Science and Business Media LLC. - 0003-9527 .- 1432-0673. ; 198:1, s. 125-187
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Tidskriftsartikel (refereegranskat)abstract
- We consider the Boltzmann equation for a gas in a horizontal slab, subject to a gravitational force. The boundary conditions are of diffusive type, specifying the wall temperatures, so that the top temperature is lower than the bottom one (Benard setup). We consider a 2-dimensional convective stationary solution, which for small Knudsen numbers is close to the convective stationary solution of the Oberbeck-Boussinesq equations, near and above the bifurcation point, and prove its stability under 2-d small perturbations, for Rayleigh numbers above and close to the bifurcation point and for small Knudsen numbers.
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