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By
Juutinen, Petri
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4 Citations
We prove a concavity maximum principle for the viscosity solutions of certain fully nonlinear and singular elliptic and parabolic partial differential equations. Our results parallel and extend those obtained by Korevaar and Kennington for classical solutions of quasilinear equations. Applications are given in the case of the singular infinity Laplace operator.
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By
Jha, Navnit; Gopal, Venu; Singh, Bhagat
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In this article, a family of high order accurate compact finite difference scheme for obtaining the approximate solution values of mildly nonlinear elliptic boundary value problems in threespace dimensions has been developed. The discretization formula is developed on a nonuniform meshes, which helps in resolving boundary and/or interior layers. The scheme involves 27 points single computational cell to achieve high order truncation errors. The proposed scheme has been applied to solve convection–diffusion equation, Helmholtz equation and nonlinear Poisson’s equation. A detailed convergence theory for the new compact scheme has been proposed using irreducible and monotone property of the iteration matrix. Numerical results show that the new compact scheme exhibit better performance in terms of
$$l^\infty $$
 and
$$l^2$$
error of the exact and approximate solution values.
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By
Ma, Li; Cheng, Liang
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3 Citations
In this paper, we study two kinds of L^{2} norm preserved nonlocal heat flows on closed manifolds. We first study the global existence, stability, and asymptotic behavior of such nonlocal heat flows. Next we give the gradient estimates of positive solutions to these heat flows.
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By
Brandolini, B.; Nitsch, C.; Salani, P.; Trombetti, C.
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8 Citations
Recently the symmetry of solutions to overdetermined problems has been established for the class of Hessian operators, including the MongeAmpère operator. In this paper we prove that the radial symmetry of the domain and of the solution to an overdetermined Dirichlet problem for the MongeAmpère equation is stable under suitable perturbations of the data.
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By
Wang, Jixiu
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In this paper, we study the existence of semiclassical states for some pLaplacian equation. Under given conditions and minimax methods, we show that this problem has at least one positive solution provided that ε ≤ E; for any m ∈ ℕ, it has m pairs solutions if ε ≤ E_{m}, where E, E_{m} are sufficiently small positive numbers. Moreover, these solutions are closed to zero in W^{1,p}(ℝ^{N}) as ε → 0.
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By
Gazzola, Filippo; Grunau, HansChristoph
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48 Citations
We prove existence and uniqueness (up to rescaling) of positive radial entire solutions of supercritical semilinear biharmonic equations. The proof is performed with a shooting method which uses the value of the second derivative at the origin as a parameter. This method also enables us to find finite time blow up solutions. Finally, we study the convergence at infinity of smooth solutions towards the explicitly known singular solution. It turns out that the convergence is different in space dimensions n ≤ 12 and n ≥ 13.
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By
Vitolo, Antonio
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In this paper we try to generalize a well known result due to Brezis on the existence of weak solutions in the whole space to secondorder fully nonlinear equations with an absorption term satisfying a Keller–Osserman condition plus an additive external source without growth condition at infinity. We also discuss constant sign solutions.
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By
Diening, Lars; Kreuzer, Christian; Schwarzacher, Sebastian
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3 Citations
The convex hull property is the natural generalization of maximum principles from scalar to vector valued functions. Maximum principles for finite element approximations are often crucial for the preservation of qualitative properties of the respective physical model. In this work we develop a convex hull property for
$$\mathbb{P }_1$$
conforming finite elements on simplicial nonobtuse meshes. The proof does not resort to linear structures of partial differential equations but directly addresses properties of the minimiser of a convex energy functional. Therefore, the result holds for very general nonlinear partial differential equations including e.g. the
$$p$$
Laplacian and the mean curvature problem. In the case of scalar equations the introduce techniques can be used to prove standard discrete maximum principles for nonlinear problems. We conclude by proving a strong discrete convex hull property on strictly acute triangulations.
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By
Kim, YongCheol; Lee, KiAhm
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5 Citations
In this paper, we consider fully nonlinear integrodifferential equations with possibly nonsymmetric kernels. We are able to find different versions of Alexandroff–Backelman–Pucci estimate corresponding to the full class
$${\mathcal {S}^{\mathfrak {L}_0}}$$
of uniformly elliptic nonlinear equations with 1 < σ < 2 (subcritical case) and to their subclass
$${\mathcal {S}_{\eta}^{\mathfrak {L}_0}}$$
with 0 < σ ≤ 1. We show that
$${\mathcal {S}_{\eta}^{\mathfrak {L}_0}}$$
still includes a large number of nonlinear operators as well as linear operators. And we show a Harnack inequality, Hölder regularity, and C^{1,α}regularity of the solutions by obtaining decay estimates of their level sets in each cases.
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