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Let Ω be a C 2 bounded domain in R N ( N ≥ 3 ), δ ( x ) = dist ( x , ∂ Ω ) and C H ( Ω ) be the best constant in the Hardy inequality with respect to Ω. We investigate positive solutions to a boundary value problem for Lane-Emden equations with Hardy potential of the form − Δ u − μ δ 2 u = u p in Ω , u = ρ ν on ∂ Ω , ( P ρ ) where 0 μ C H ( Ω ) , ρ is a positive parameter, ν is a positive Radon measure on ∂Ω with norm 1 and 1 p N μ , with N μ being a critical exponent depending on N and μ. It is known from [22] that there exists a threshold value ρ ⁎ such that problem ( P ρ ) admits a positive solution if 0 ρ ≤ ρ ⁎ , and no positive solution if ρ > ρ ⁎ . In this paper, we go further in the study of the solution set of ( P ρ ) . We show that the problem admits at least two positive solutions if 0 ρ ρ ⁎ and a unique positive solution if ρ = ρ ⁎ . We also prove the existence of at least two positive solutions for Lane-Emden systems { − Δ u − μ δ 2 u = v p in Ω , − Δ v − μ δ 2 v = u q in Ω , u = ρ ν , v = σ τ on ∂ Ω , under the smallness condition on the positive parameters ρ and σ.

We study the kernel function of the operator u $\rightarrow$ L $\mu$ u = --$\Delta$u + $\mu$ |x| 2 u in a bounded smooth domain $\Omega$ $\subset$ R N + such that 0 $\in$ $\partial$$\Omega$, where $\mu$ $\ge$ -- N 2 4 is a constant. We show the existence of a Poisson kernel vanishing at 0 and a singular kernel with a singularity at 0. We prove the existence and uniqueness of weak solutions of L $\mu$ u = 0 in $\Omega$ with boundary data $\nu$ + k$\delta$ 0 , where $\nu$ is a Radon measure on $\partial$$\Omega$ \ {0}, k $\in$ R and show that this boundary data corresponds in a unique way to the boundary trace of positive solution of L $\mu$ u = 0 in $\Omega$.

The aim of the paper is to study a gradient constrained problem associated with a linear operator. Two types of problems are investigated. The first one is the equivalence between a non-constant gradient constrained problem and a suitable obstacle problem, where the obstacle solves a Hamilton-Jacobi equation in the viscosity sense. The equivalence result is obtained under a condition on the gradient constraint. The second problem is the existence of Lagrange multipliers. We prove that the non-constant gradient constrained problem admits a Lagrange multiplier, which is a Radon measure if the free term of the equation f ∈ L p , p > 1 . If f is a positive constant, we regularize the result, namely we prove that the Lagrange multipliers belong to L 2 .

In this work, we consider the one dimensional very singular fourth-order equation for solid-on-solid model in attachment-detachment-limit regime with exponential nonlinearity $$h_t = \nabla \cdot (\frac{1}{|\nabla h|} \nabla e^{\frac{\delta E}{\delta h}}) =\nabla \cdot (\frac{1}{|\nabla h|}\nabla e^{- \nabla \cdot (\frac{\nabla h}{|\nabla h|})})$$ where total energy $E=\int |\nabla h|$ is the total variation of $h$. Using a logarithmic correction $E=\int |\nabla h|\ln|\nabla h| d x$ and gradient flow structure with a suitable defined functional, we prove the evolution variational inequality solution preserves a positive gradient $h_x$ which has upper and lower bounds but in BV space. We also obtain the global strong solution to the solid-on-solid model which allows an asymmetric singularity $h_{xx}^+$ happens., Comment: 15 pages

In this article we consider weak solutions of the Euler-α equations in the full plane. We take, as initial unfiltered vorticity, an arbitrary nonnegative, compactly supported, bounded Radon measure. Global well-posedness for the corresponding initial value problem is due M. Oliver and S. Shkoller. We show that, for all time, the support of the unfiltered vorticity is contained in a disk whose radius grows no faster than O ( ( t log ⁡ t ) 1 / 4 ) . This result is an adaptation of the corresponding result for the incompressible 2D Euler equations with initial vorticity compactly supported, nonnegative, and p-th power integrable, p > 2 , due to D. Iftimie, T. Sideris and P. Gamblin and, independently, to Ph. Serfati.

We study the existence of solutions of the semilinear parabolic equation ∂ t u − Δ u = u p + Λ in D ′ ( R N × I ) . Here N ≥ 2 , p > 1 , ( N − 2 ) p N and I ⊂ R is an open interval. It is assumed that Λ ∈ D ′ ( R N × I ) is supported on { ( ξ ( t ) , t ) ∈ R N + 1 ; t ∈ I } for some ξ ∈ C 1 / 2 ( I ‾ ; R N ) and is given by Λ ( φ ) = ∫ I φ ( ξ ( t ) , t ) d μ ( t ) , where μ is a Radon measure on I satisfying μ ( ( a , b ) ) ≤ ϕ ( b − a ) ( a , b ∈ I ) for some positive function ϕ ( η ) . In the case where I is bounded, we give conditions on the behavior of ϕ ( η ) near η = 0 for the existence and nonexistence of solutions. The existence for I = R is also shown under additional conditions on p and the behavior of ϕ ( η ) near η = ∞ . Moreover, the singularity of solutions at x = ξ ( t ) is examined.

The paper deals with stationary equation governed by the operator − ∇ ⋅ A ( x , ∇ u ) = μ in the case where A ( x , ξ ) is a maximal monotone graph and μ is a Radon measure. Our main interest concerns the typical situation where A ( x , . ) is defined only in a bounded region of I R n ; so that A ( x , . ) does not satisfy the standard polynomial growth control condition.

We study the existence of weak solutions to (E) ( − Δ ) α u + g ( u ) = ν in a bounded regular domain Ω in R N ( N ≥ 2 ) which vanish in R N ∖ Ω , where ( − Δ ) α denotes the fractional Laplacian with α ∈ ( 0 , 1 ) , ν is a Radon measure and g is a nondecreasing function satisfying some extra hypotheses. When g satisfies a subcritical integrability condition, we prove the existence and uniqueness of weak solution for problem (E) for any measure. In the case where ν is a Dirac measure, we characterize the asymptotic behavior of the solution. When g ( r ) = | r | k − 1 r with k supercritical, we show that a condition of absolute continuity of the measure with respect to some Bessel capacity is a necessary and sufficient condition in order (E) to be solved.

The paper deals with Monge–Kantorovich equation (MK for short) in an open bounded domain Ω with Dirichlet boundary condition. We study existence and uniqueness of a solution to the associated evolution problem (EMK for short) and we prove the convergence to a solution of MK, when time goes to ∞. A solution is a couple ( u , Φ ) , where u is the potential and Φ is the transportation flux. We study the problem for a given Radon measure source term and we show how to use the numerical method of Dumont and Igbida (2009) [22] to provide numerical approximation of the solution of MK.

In this paper we study the Cauchy problem for the singular evolution p-Laplacian equations with gradient term and source on the assumption of measures as initial conditions. For the supercritical case q > p − 1 + p / N , we obtain that for every nonnegative solution there exists a nonnegative Radon measure μ as initial trace and μ has some local regularity.

The conical differentiability of solutions to the parabolic variational inequality with respect to the right-hand side is proved in the paper. From one side the result is based on the Lipschitz continuity in H 1 2 ,1 (Q) of solutions to the variational inequality with respect to the right-hand side. On the other side, in view of the polyhedricity of the convex cone K={v∈ H ;v |Σ c ⩾0,v |Σ d =0}, we prove new results on sensitivity analysis of parabolic variational inequalities. Therefore, we have a positive answer to the question raised by Fulbert Mignot (J. Funct. Anal. 22 (1976) 25–32).

We prove some partial regularity results for the entropy solution u of the so-called relativistic heat equation. In particular, under some assumptions on the initial condition u0, we prove that ut(t) is a Radon measure in RN. Moreover, if u0 is log-concave inside its support Ω, Ω being a convex set, then we show the solution u(t) is also log-concave in its support Ω(t). This implies its smoothness in Ω(t). In that case we can give a simpler characterization of the notion of entropy solution.

We introduce a new space of Morrey type in W − 1, q (Ω) denoted M λ −1, q (Ω) . Then, we prove the following optimal condition: if u is a local solution of for some λ > N − p, 1 p + 1 q = 1 if and only if for all subsets Ω′ relatively compact, there exist C > 0, and σ > N − p such that for all x in Ω′, all R > 0: B(x, 2R) ⊂ Ω′ , we have (H) ∫ B(x,R ¦▽u¦ p dy ⩽ C · R σ . In particular, this result implies that any locally bounded solution of ( P 1) is locally Holder continuous, provided that T belongs to Mλ−1,q, for λ > N − p. Since we need boundness of solutions, we prove in the second section the boundness property of the Dirichlet problem associated to ( P 1) for a large class of T (including all the right hand side T considered in the literature). The method relies on some property of Radon measure in W− 1,q.

We consider parabolic equations of the type u t − div A ( x , t , D u ) = μ having a Radon measure on the right-hand side and prove fractional integrability and differentiability results of Calderon–Zygmund type for weak solutions. We extend some of the integrability results for elliptic equations achieved by G. Mingione (2007) [24] to the parabolic setting and locally recover the integrability results of L. Boccardo, A. DallʼAglio, T. Gallouet, and L. Orsina (1997) in [5] .

We study nonnegative solutions of the filtration equation u t =Δ ϕ ( u ) in R N (N⩾2) , where ϕ is continuous, increasing and sublinear. More precisely, we study the Razor Blades, i.e., solutions which may be singular at | x |=0 for t >0, and start with zero initial data. We first prove a nonexistence result when ϕ is too sublinear and we show an axial trace result in the other case: there exist a time τ ≡ τ ( u ) and a Radon measure λ on [0, τ ) such that u t = Δ ϕ(u)+δ 0 ⊗λ(t) in D ′( R N ×[0,τ)). Then u has a strong singularity on | x |=0 for any t > τ . We then prove existence of such solutions for any λ and τ as above, and give a uniqueness result for those solutions. Finally, we make a complete study of self-similar solutions (in the power case) which classify the possible asymptotic behaviours.