662 research outputs found

    Uniqueness of solutions of semilinear Poisson equations

    Full text link

    Nonlinear Hodge maps

    Full text link
    We consider maps between Riemannian manifolds in which the map is a stationary point of the nonlinear Hodge energy. The variational equations of this functional form a quasilinear, nondiagonal, nonuniformly elliptic system which models certain kinds of compressible flow. Conditions are found under which singular sets of prescribed dimension cannot occur. Various degrees of smoothness are proven for the sonic limit, high-dimensional flow, and flow having nonzero vorticity. The gradient flow of solutions is estimated. Implications for other quasilinear field theories are suggested.Comment: Slightly modified and updated version; tcilatex, 32 page

    A Fresh Look at Entropy and the Second Law of Thermodynamics

    Full text link
    This paper is a non-technical, informal presentation of our theory of the second law of thermodynamics as a law that is independent of statistical mechanics and that is derivable solely from certain simple assumptions about adiabatic processes for macroscopic systems. It is not necessary to assume a-priori concepts such as "heat", "hot and cold", "temperature". These are derivable from entropy, whose existence we derive from the basic assumptions. See cond-mat/9708200 and math-ph/9805005.Comment: LaTex file. To appear in the April 2000 issue of PHYSICS TODA

    Conditional regularity of solutions of the three dimensional Navier-Stokes equations and implications for intermittency

    Full text link
    Two unusual time-integral conditional regularity results are presented for the three-dimensional Navier-Stokes equations. The ideas are based on L2mL^{2m}-norms of the vorticity, denoted by Ωm(t)\Omega_{m}(t), and particularly on Dm=ΩmαmD_{m} = \Omega_{m}^{\alpha_{m}}, where αm=2m/(4m3)\alpha_{m} = 2m/(4m-3) for m1m\geq 1. The first result, more appropriate for the unforced case, can be stated simply : if there exists an 1m<1\leq m < \infty for which the integral condition is satisfied (Zm=Dm+1/DmZ_{m}=D_{m+1}/D_{m}) 0tln(1+Zmc4,m)dτ0 \int_{0}^{t}\ln (\frac{1 + Z_{m}}{c_{4,m}}) d\tau \geq 0 then no singularity can occur on [0,t][0, t]. The constant c4,m2c_{4,m} \searrow 2 for large mm. Secondly, for the forced case, by imposing a critical \textit{lower} bound on 0tDmdτ\int_{0}^{t}D_{m} d\tau, no singularity can occur in Dm(t)D_{m}(t) for \textit{large} initial data. Movement across this critical lower bound shows how solutions can behave intermittently, in analogy with a relaxation oscillator. Potential singularities that drive 0tDmdτ\int_{0}^{t}D_{m} d\tau over this critical value can be ruled out whereas other types cannot.Comment: A frequency was missing in the definition of D_{m} in (I5) v3. 11 pages, 1 figur

    Existence, Regularity, and Properties of Generalized Apparent Horizons

    Full text link
    We prove a conjecture of Tom Ilmanen's and Hubert Bray's regarding the existence of the outermost generalized apparent horizon in an initial data set and that it is outer area minimizing.Comment: 16 pages, thoroughly revised, no major changes, to appear in Comm. Math. Phy

    Supercritical biharmonic equations with power-type nonlinearity

    Full text link
    The biharmonic supercritical equation Δ2u=up1u\Delta^2u=|u|^{p-1}u, where n>4n>4 and p>(n+4)/(n4)p>(n+4)/(n-4), is studied in the whole space Rn\mathbb{R}^n as well as in a modified form with λ(1+u)p\lambda(1+u)^p as right-hand-side with an additional eigenvalue parameter λ>0\lambda>0 in the unit ball, in the latter case together with Dirichlet boundary conditions. As for entire regular radial solutions we prove oscillatory behaviour around the explicitly known radial {\it singular} solution, provided p((n+4)/(n4),pc)p\in((n+4)/(n-4),p_c), where pc((n+4)/(n4),]p_c\in ((n+4)/(n-4),\infty] is a further critical exponent, which was introduced in a recent work by Gazzola and the second author. The third author proved already that these oscillations do not occur in the complementing case, where ppcp\ge p_c. Concerning the Dirichlet problem we prove existence of at least one singular solution with corresponding eigenvalue parameter. Moreover, for the extremal solution in the bifurcation diagram for this nonlinear biharmonic eigenvalue problem, we prove smoothness as long as p((n+4)/(n4),pc)p\in((n+4)/(n-4),p_c)

    Continuous and discrete Clebsch variational principles

    Full text link
    The Clebsch method provides a unifying approach for deriving variational principles for continuous and discrete dynamical systems where elements of a vector space are used to control dynamics on the cotangent bundle of a Lie group \emph{via} a velocity map. This paper proves a reduction theorem which states that the canonical variables on the Lie group can be eliminated, if and only if the velocity map is a Lie algebra action, thereby producing the Euler-Poincar\'e (EP) equation for the vector space variables. In this case, the map from the canonical variables on the Lie group to the vector space is the standard momentum map defined using the diamond operator. We apply the Clebsch method in examples of the rotating rigid body and the incompressible Euler equations. Along the way, we explain how singular solutions of the EP equation for the diffeomorphism group (EPDiff) arise as momentum maps in the Clebsch approach. In the case of finite dimensional Lie groups, the Clebsch variational principle is discretised to produce a variational integrator for the dynamical system. We obtain a discrete map from which the variables on the cotangent bundle of a Lie group may be eliminated to produce a discrete EP equation for elements of the vector space. We give an integrator for the rotating rigid body as an example. We also briefly discuss how to discretise infinite-dimensional Clebsch systems, so as to produce conservative numerical methods for fluid dynamics

    Pontryagin´s principle for state-constrained boundary control problems of semilinear parabolic equations

    Get PDF
    This paper deals with state-constrained optimal control problems governed by semilinear parabolic equations. We establish a minimum principle of Pontryagin's type. To deal with the state constraints, we introduce a penalty problem by using Ekeland's principle. The key tool for the proof is the use of a special kind of spike perturbations distributed in the domain where the controls are de ned. Conditions for normality of optimality conditions are given

    A remark on an overdetermined problem in Riemannian Geometry

    Full text link
    Let (M,g)(M,g) be a Riemannian manifold with a distinguished point OO and assume that the geodesic distance dd from OO is an isoparametric function. Let ΩM\Omega\subset M be a bounded domain, with OΩO \in \Omega, and consider the problem Δpu=1\Delta_p u = -1 in Ω\Omega with u=0u=0 on Ω\partial \Omega, where Δp\Delta_p is the pp-Laplacian of gg. We prove that if the normal derivative νu\partial_{\nu}u of uu along the boundary of Ω\Omega is a function of dd satisfying suitable conditions, then Ω\Omega must be a geodesic ball. In particular, our result applies to open balls of Rn\mathbb{R}^n equipped with a rotationally symmetric metric of the form g=dt2+ρ2(t)gSg=dt^2+\rho^2(t)\,g_S, where gSg_S is the standard metric of the sphere.Comment: 8 pages. This paper has been written for possible publication in a special volume dedicated to the conference "Geometric Properties for Parabolic and Elliptic PDE's. 4th Italian-Japanese Workshop", organized in Palinuro in May 201
    corecore