Resolutions of the prescribed volume form equation

For a given volume formfdx on a bounded regular domain in IR ⁿ , we are looking for a transformationu of , keeping the boundary fixed and which sends the Lebesgue measuredx intofdx (i.e. we solve det (u)=f. Forf in various spaces, we propose two different constructions which ensure the existence ofu with some gain of regularity. Our methods permit the recovery Dacorogna and Moser's results [4], but also, we prove the existence of suchu in Hölder spaces forf inC ⁰, or even inL .     

Global attractors for p-Laplacian equation

The existence of a -global attractor is proved for the p-Laplacian equation ut−div(|∇u|^p−2∇u)+f(u)=g on a bounded domain Ω⊂Rn(n?3) with Dirichlet boundary condition, where p?2. The nonlinear term f is supposed to satisfy the polynomial growth condition of arbitrary order c₁q|u|−k?f(u)u?c₂q|u|+k and f^′(u)?−l, where q?2 is arbitrary. There is no other restriction on p and q. The asymptotic compactness of the corresponding semigroup is proved by using a new a priori estimate method, called asymptotic a priori estimate.     

Abschätzungen der unvollständigen Beta-Funktion in mehreren reellen Variablen und Parametern

The integral ofu ₁ ^p–1 ...u _n ^pn–1 (1–u₁–...–u_n)^q, extended over then-simplex {(u ₁,...,u _n) ₊ ⁿ :u ₁/x ₁+...+u _n/x _n<1}, is estimated from below under the condition thatp ₁+...+p_n+qc=p₁/x₁+...+p_n/x_n=const.     

Existence Theorems for Certain Elliptic and Parabolic Semilinear Equations

Existence theorems for the nonlinear parabolic differential equation −∂u/∂t + Δu + |u|^p + f(x, t) = 0 in ⁿ × [0, ∞) with zero initial value are established given explicit conditions on the nonhomogeneous termf(x, t). An existence theorem is also demonstrated for the corresponding elliptic equation.     

Finite‐dimensional attractors and exponential attractors for degenerate doubly nonlinear equations

We consider the following doubly nonlinear parabolic equation in a bounded domain Ω??³: where the nonlinearity f is allowed to have a degeneracy with respect to ?_tu of the form ?_tu|?_tu|^p at some points x∈Ω. Under some natural assumptions on the nonlinearities f and g, we prove the existence and uniqueness of a solution of that problem and establish the finite‐dimensionality of global and exponential attractors of the semigroup associated with this equation in the appropriate phase space. Copyright © 2009 John Wiley & Sons, Ltd.     

Nonexistence of Ground States of -△u = u^p - u^q

The authors use the method of moving spheres to prove the nonexistence of ground states of -△u = u^p - u^q for n≥3,-∞〈p〈（n＋2）/（n-2） and q〉max （1,p）,
In fact this conclusion is a special case of -△u =f（u） for n≥2.     

Large solutions of elliptic equations with a weakly superlinear nonlinearity

This paper studies the asymptotic behavior near the boundary for large solutions of the semilinear equation Δu + au = b(x)f(u) in a smooth bounded domain Ω of ℝ^N with N ≥ 2, where a is a real parameter and b is a nonnegative smooth function on . We assume that f(u) behaves like u(ln u)^α as u → ∞, for some α > 2. It turns out that this case is more difficult to handle than those where f(u) grows like u ^p (p > 1) or faster at infinity. Under suitable conditions on the weight function b(x), which may vanish on ∂Ω, we obtain the first order expansion of the large solutions near the boundary. We also obtain some uniqueness results. Research of both authors supported by the Australian Research Council.     

Very weak solutions to elliptic equations with nonlinear Neumann boundary conditions

Consider the equation −Δu = 0 in a bounded smooth domain , complemented by the nonlinear Neumann boundary condition ∂_ν u = f(x, u) − u on ∂Ω. We show that any very weak solution of this problem belongs to L ^∞(Ω) provided f satisfies the growth condition |f(x, s)| ≤ C(1 + |s| ^p ) for some p ∈ (1, p*), where . If, in addition, f(x, s) ≥ −C + λs for some λ > 1, then all positive very weak solutions are uniformly a priori bounded. We also show by means of examples that p* is a sharp critical exponent. In particular, using variational methods we prove the following multiplicity result: if N ∈ {3, 4} and f(x, s) =  s ^p then there exists a domain Ω and such that our problem possesses at least two positive, unbounded, very weak solutions blowing up at a prescribed point of ∂Ω provided . Our regularity results and a priori bounds for positive very weak solutions remain true if the right-hand side in the differential equation is of the form h(x, u) with h satisfying suitable growth conditions.     

Existence and Asymptotic Behaviour of Large Solutions of Semilinear Elliptic Equations

In this paper we are interested in the semilinear elliptic equations of the type u=u(,u), on bounded smooth domain of R ⁿ . We also treat existence of positive solution of u=p(x)f(u), which explodes near the boundary of (called large solutions). Our approach is based on potential theory.     

A Bahri–Lions theorem revisited

In 1988, A. Bahri and P.L. Lions [A. Bahri, P.L. Lions, Morse-index of some min–max critical points. I. Application to multiplicity results, Comm. Pure Appl. Math. 41 (1988) 1027–1037] studied the following elliptic problem:

where Ω is a bounded smooth domain of , 2<p<(2N−2)/(N−2) and f(x,u) is not assumed to be odd in u. They proved the existence of infinitely many solutions under an appropriate growth restriction on f. In the present paper, we improve this result by showing that under the same growth assumption on f the problem admits in fact infinitely many sign-changing solutions. In addition we derive an estimate on the number of their nodal domains. We also deal with the corresponding fourth order equation Δ²u=|u|^p−2u+f(x,u) with both Dirichlet and Navier boundary conditions, as well as with strongly coupled elliptic systems.     

Existence and nonexistence results for quasilinear elliptic equations involving the <Emphasis Type="Italic">p</Emphasis>-Laplacian with a critical potential

This paper deals with the existence and nonexistence results for quasilinear elliptic equations of the form -_pu=f(x, u), where _p:=div(|u|^p-2u), p>1, and the solutions are understood in the sense of renormalized or, equivalently, entropy solutions. In particular we prove nonexistence results in the case f(x,u)=u^p|x|^-p, that is related to a classical Hardy inequality. Mathematics Subject Classification (2000) 35D05, 35D10, 35J20, 35J25, 35J70     

A sharp form of Poincaré type inequalities on balls and spheres

Summary Letu be a real valued function on ann-dimensional Riemannian manifoldM ⁿ. We consider an inequality between theL ^q-norm ofu minus its mean value overM ⁿ and theL ^p-norm of the gradient ofu.The best constant in such inequality is exhibited in the following cases: i)M ⁿ is an open ball inIR ⁿ andp=1, 0<qn/(n–1); ii)M ⁿ is a sphere inIR ⁿ +1 and eitherp=1, 0<qn/(n–1) orp>n,q=.

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Sunto Siau una funzione a valori reali dafinita su una varietà riemannianan-dimensionaleM ⁿ. Si considera una disuguaglianza tra la normaL ^q diu meno il suo valor medio suM ⁿ e la normaL ^p del gradiente diu.Si determina la costante ottimale in tale disuguaglianza nei seguenti casi: i)M ⁿ è un disco aperto inIR ⁿ ep=1, 0<qn/(n–1); ii)M ⁿ è una sfera inIR ⁿ +1 ep=1, 0<qn/(n–1) oppurep>n,q=.

     

Sharp estimates for\bar \partial on convex domains of finite typeon convex domains of finite type

Let Ω be a bounded convex domain in C _n , with smooth boundary of finite typem. The equation is solved in Ω with sharp estimates: iff has bounded coefficients, the coefficients of our solutionu are in the Lipschitz space Λ. Optimal estimates are also given when data have coefficients belonging toL ^p(Ω),p≥1. We solve the -equation by means of integral operators whose kernels are not based on the choice of a “good” support function. Weighted kernels are used; in order to reflect the geometry ofbΩ, we introduce a weight expressed in terms of the Bergman kernel of Ω.     

Lower Bound for the Poles of Igusa’s p-adic Zeta Functions

Let K be a p-adic field, R the valuation ring of K, P the maximal ideal of R and q the cardinality of the residue field R/P. Let f be a polynomial over R in n >1 variables and let χ be a character of . Let M _i (u) be the number of solutions of f = u in (R/P ⁱ ) ⁿ for and. These numbers are related with Igusa’s p-adic zeta function Z _f,χ(s) of f. We explain the connection between the M _i (u) and the smallest real part of a pole of Z _f,χ(s). We also prove that M _i (u) is divisible by , where the corners indicate that we have to round up. This will imply our main result: Z _f,χ(s) has no poles with real part less than − n/2. We will also consider arbitrary K-analytic functions f.     

One Boundary Version of Morera's Theorem

Let D be a bounded domain in C ⁿ (n>1) with a connected smooth boundary D and let f be a continuous function on D. We consider conditions (generalizing those of the Hartogs–Bochner theorem) for holomorphic extendability of f to D. As a corollary we derive some boundary analog of Morera's theorem claiming that if the integrals of f vanish over the intersection of the boundary of the domain with complex curves in some class then f extends holomorphically to the domain.     

The minimality of the map \frac{x}{\|{x}\|} for weighted energy

In this paper, we investigate the minimality of the map from the Euclidean unit ball Bⁿ to its boundary 핊ⁿ⁻¹ for weighted energy functionals of the type E_p,f = ∫_Bⁿ f(r)‖∇ u‖^p dx, where f is a non-negative function. We prove that in each of the two following cases:

Some Optimization Problems for <Emphasis Type="Italic">p</Emphasis>-Laplacian Type Equations

In this paper we study some optimization problems for nonlinear elastic membranes. More precisely, we consider the problem of optimizing the cost functional over some admissible class of loads f where u is the (unique) solution to the problem −Δ _p u+|u| ^p−2 u=0 in Ω with |∇ u| ^p−2 u _ν =f on ∂Ω. Supported by Universidad de Buenos Aires under grant X078, by ANPCyT PICT No. 2006-290 and CONICET (Argentina) PIP 5478/1438. J. Fernández Bonder is a member of CONICET. Leandro M. Del Pezzo is a fellow of CONICET.     

Isolated singularities of solutions to quasi-linear elliptic equations with absorption

We study the problem of removability of isolated singularities for a general second-order quasi-linear equation in divergence form −divA(x,u,∇u)+a₀(x,u)+g(x,u)=0 in a punctured domain Ω?{0}, where Ω is a domain in Rn, n?3. The model example is the equation −Δpu+gu|u|^p−2+u|u|^q−1=0, q>p−1>0, p<n. Assuming that the lower-order terms satisfy certain non-linear Kato-type conditions, we prove that for all point singularities of the above equation are removable, thus extending the seminal result of Brezis and Véron.