Finite volume element method for monotone nonlinear elliptic problems

In this article, we consider the finite volume element method for the monotone nonlinear second‐order elliptic boundary value problems. With the assumptions which guarantee that the corresponding operator is strongly monotone and Lipschitz‐continuous, and with the minimal regularity assumption on the exact solution, that is, u∈H¹(Ω), we show that the finite volume element method has a unique solution, and the finite volume element approximation is uniformly convergent with respect to the H¹ ‐norm. If u∈H^1+ε(Ω),0 < ε ≤ 1, we develop the optimal convergence rate \begin{align*}\mathcal{O}(h^{\epsilon})\end{align*} in the H¹ ‐norm. Moreover, we propose a natural and computationally easy residual‐based H¹ ‐norm a posteriori error estimator and establish the global upper bound and local lower bounds on the error. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Nonlinear Boundary Value Problems for Second Order Differential Inclusions

In this paper we study two boundary value problems for second order strongly nonlinear differential inclusions involving a maximal monotone term. The first is a vector problem with Dirichlet boundary conditions and a nonlinear differential operator of the form x ↦ a(x, x′)′. In this problem the maximal monotone term is required to be defined everywhere in the state space ℝ^N. The second problem is a scalar problem with periodic boundary conditions and a differential operator of the form x ↦ (a(x)x′)′. In this case the maximal monotone term need not be defined everywhere, incorporating into our framework differential variational inequalities. Using techniques from multivalued analysis and from nonlinear analysis, we prove the existence of solutions for both problems under convexity and nonconvexity conditions on the multivalued right-hand side.     

Convergence rates in regularization for the case of monotone perturbations

Convergence rates are justified for regularized solutions of a Hammerstein operator equation of the form x + F ₂ F ₁(x) = f in the Banach space with monotone perturbations f ₂ ^h and f ₁ ^h .     

Solution of monotone nonlinear elliptic boundary value problems

LetL be a linear uniformly elliptic second order operator. The boundary value problem is solved for the nonlinear elliptic equationLu=f(x, u) wheref(x, u) is a monotone increasing function ofu for each pointx in the domain. A descent technique based on Newton's method is shown to yield a sequence of iterates which converges uniformly and quadratically to the solution. The convergence is independent of the choice for the initial iterate. Numerical results in two dimensions are presented.     

On some operator monotone functions

We try to find a continuous functionu defined on a real right half-line with the range (0, ) such thatu ^–1 is operator monotone. We then look for another functionv such thatv(u ^–1) is operator monotone, namely,u(A)u(B) impliesv(A)v(B) for self-adjoint operatorsA andB.     

Parabolic Itô equations with monotone nonlinearities

In this paper the equation u_t = Lu ? F(u) + α(t, ω) is studied, where u(t) ?B₀ a Banach space. L is an unbounded self-adjoint negative definite operator. F is a monotone nonlinear potential operator. α(t, ω) is a white noise process on B₀. With suitable further restrictions on L and F it is proved that the equation has a unique solution. As t → ∞ the distribution of u(t, ω) approaches a stationary distribution which is calculated explicitly.     

A monotone iterative technique for stationary and time dependent problems in Banach spaces

An abstract monotone iterative method is developed for operators between partially ordered Banach spaces for the nonlinear problem Lu=Nu and the nonlinear time dependent problem u^′=(L+N)u. Under appropriate assumptions on L and N we obtain maximal and minimal solutions as limits of monotone sequences of solutions of linear problems. The results are illustrated by means of concrete examples.     

Generalized equations,variational inequalities and a weak Kantorovich theorem

We prove a Kantorovich-type theorem on the existence and uniqueness of the solution of a generalized equation of the form f(u)+g(u) ' 0f(u)+g(u)\owns 0 where f is a Fréchet-differentiable function and g is a maximal monotone operator defined on a Hilbert space. The depth and scope of this theorem is such that when we specialize it to nonlinear operator equations, variational inequalities and nonlinear complementarity problems we obtain novel results for these problems as well. Our approach to the solution of a generalized equation is iterative, and the solution is obtained as the limit of the solutions of partially linearized generalized Newton subproblems of the type Az+g(z) ' bAz+g(z)\owns b where A is a linear operator.     

Norm inequalities related to operator monotone functions

Let A,B be positive semidefinite matrices and any unitarily invariant norm on the space of matrices. We show for any non-negative operator monotone function f(t) on , and for non-negative increasing function g(t) on with g(0) = 0 and , whose inverse function is operator monotone. Received: 1 February 1999     

The conformal plate buckling equation

The linear equation Δ²u = 1 for the infinitesimal buckling under uniform unit load of a thin elastic plate over ?² has the particularly interesting nonlinear generalization Δ_g²u = 1, where Δ_g = e^?2u Δ is the Laplace‐Beltrami operator for the metric g = e^2ug₀, with g₀ the standard Euclidean metric on ?². This conformal elliptic PDE of fourth order is equivalent to the nonlinear system of elliptic PDEs of second order Δu(x)+K_g(x) exp(2u(x)) = 0 and Δ K_g(x) + exp(2u(x)) = 0, with x ∈ ?², describing a conformally flat surface with a Gauss curvature function K_g that is generated self‐consistently through the metric's conformal factor. We study this conformal plate buckling equation under the hypotheses of finite integral curvature ∫ K_g exp(2u)dx = κ, finite area ∫ exp(2u)dx = α, and the mild compactness condition K₊ ∈ L¹(B₁(y)), uniformly w.r.t. y ∈ ?². We show that asymptotically for |x|→∞ all solutions behave like u(x) = ?(κ/2π)ln |x| + C + o(1) and K(x) = ?(α/2π) ln|x| + C + o(1), with κ ∈ (2π, 4π) and . We also show that for each κ ∈ (2π, 4π) there exists a K^* and a radially symmetric solution pair u, K, satisfying K(u) = κ and maxK = K^*, which is unique modulo translation of the origin, and scaling of x coupled with a translation of u. © 2001 John Wiley & Sons, Inc.     

A discrepancy principle for equations with monotone continuous operators

A discrepancy principle for solving nonlinear equations with monotone operators given noisy data is formulated. The existence and uniqueness of the corresponding regularization parameter a(δ) are proved. Convergence of the solution obtained by the discrepancy principle is justified. The results are obtained under natural assumptions on the nonlinear operator.     

Induced path transit function, monotone and Peano axioms

The induced path transit function J(u,v) in a graph consists of the set of all vertices lying on any induced path between the vertices u and v. A transit function J satisfies monotone axiom if x,y∈J(u,v) implies J(x,y)⊆J(u,v). A transit function J is said to satisfy the Peano axiom if, for any u,v,w∈V,x∈J(v,w), y∈J(u,x), there is a z∈J(u,v) such that y∈J(w,z). These two axioms are equivalent for the induced path transit function of a graph. Planar graphs for which the induced path transit function satisfies the monotone axiom are characterized by forbidden induced subgraphs.     

Solution of monotone complementarity problems with locally Lipschitzian functions

The paper deals with complementarity problems CP(F), where the underlying functionF is assumed to be locally Lipschitzian. Based on a special equivalent reformulation of CP(F) as a system of equationsφ(x)=0 or as the problem of minimizing the merit functionΘ=1/2∥Φ∥ ₂ ² , we extend results which hold for sufficiently smooth functionsF to the nonsmooth case. In particular, ifF is monotone in a neighbourhood ofx, it is proved that 0 εδθ(x) is necessary and sufficient forx to be a solution of CP(F). Moreover, for monotone functionsF, a simple derivative-free algorithm that reducesΘ is shown to possess global convergence properties. Finally, the local behaviour of a generalized Newton method is analyzed. To this end, the result by Mifflin that the composition of semismooth functions is again semismooth is extended top-order semismooth functions. Under a suitable regularity condition and ifF isp-order semismooth the generalized Newton method is shown to be locally well defined and superlinearly convergent with the order of 1+p.     

Regressions and monotone chains: A ramsey-type extremal problem for partial orders

Aregression is a functiong from a partially ordered set to itself such thatg(x)≦x for allz. Amonotone k-chain is a chain ofk elementsx ₁<x ₂ <...<x _k such thatg(x ₁)≦g(x ₂)≦...≦g(x _k ). If a partial order has sufficiently many elements compared to the size of its largest antichain, every regression on it will have a monotone (k + 1)-chain. Fixingw, letf(w, k) be the smallest number such that every regression on every partial order with size leastf(w, k) but no antichain larger thanw has a monotone (k + 1)-chain. We show thatf(w, k)=(w+1) ^k . Dedicated to Paul Erdős on his seventieth birthday Research supported in part by the National Science Foundation under ISP-80-11451.     

On a fractional partial differential equation with dominating linear part

It is proved that there is a (weak) solution of the equation u_t=a*u_xx+b*g(u_x)_x+f, on ℝ⁺ (where * denotes convolution over (−∞, t)) such that u_x is locally bounded. Emphasis is put on having the assumptions on the initial conditions as weak as possible. The kernels a and b are completely monotone and if a(t)=t^−α, b(t)=t^−β, and g(ξ)∼sign(ξ)∣ξ∣^γ for large ξ, then the main assumption is that α>(2γ+2)/(3γ+1)β+(2γ−2)/(3γ+1). © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Homogenization of monotone operators in divergence form with <Emphasis Type="Italic">x</Emphasis>-dependent multivalued graphs

In a previous paper [4], we proved the existence of solutions to −div a(x, grad u) = f , together with appropriate boundary conditions, whenever a(x, e) belongs, for every fixed x, to a certain class of maximal monotone graphs in e. Here, we derive the corresponding homogenization result, letting a(x, e) depend upon a parameter ε, and imposing adequate ε-uniform boundedness and coercivity properties. The resulting homogenized graphs belong to the same class of maximal monotone graphs. Our results do not assume any kind of periodicity.      

On the noise sensitivity of monotone functions

It is known that for all monotone functions f : {0, 1}ⁿ → {0, 1}, if x ∈ {0, 1}ⁿ is chosen uniformly at random and y is obtained from x by flipping each of the bits of x independently with probability ? = n^?α, then P[f(x) ≠ f(y)] < cn^?α+1/2, for some c > 0. Previously, the best construction of monotone functions satisfying P[f_n(x) ≠ f_n(y)] ≥ δ, where 0 < δ < 1/2, required ? ≥ c(δ)n^?α, where α = 1 ? ln 2/ln 3 = 0.36907 …, and c(δ) > 0. We improve this result by achieving for every 0 < δ < 1/2, P[f_n(x) ≠ f_n(y)] ≥ δ, with:

• ? = c(δ)n^?α for any α < 1/2, using the recursive majority function with arity k = k(α);
• ? = c(δ)n^?1/2log^tn for t = log₂ = .3257 …, using an explicit recursive majority function with increasing arities; and
• ? = c(δ)n^?1/2, nonconstructively, following a probabilistic CNF construction due to Talagrand.

We also study the problem of achieving the best dependence on δ in the case that the noise rate ? is at least a small constant; the results we obtain are tight to within logarithmic factors. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 333–350, 2003     

Comparison results for nonlinear elliptic equations with lower–order terms

We consider a solution u of the homogeneous Dirichlet problem for a class of nonlinear elliptic equations in the form A(u) + g(x, u) = f, where the principal term is a Leray–Lions operator defined on and g(x, u) is a term having the same sign as u and satisfying suitable growth assumptions. We prove that the rearrangement of u can be estimated by the solution of a problem whose data are radially symmetric.     

Eigenvalue results for pseudomonotone perturbations of maximal monotone operators

Let X be an infinite-dimensional real reflexive Banach space such that X and its dual X* are locally uniformly convex. Suppose that T: X?D(T) → 2 ^X * is a maximal monotone multi-valued operator and C: X?D(C) → X* is a generalized pseudomonotone quasibounded operator with L ? D(C), where L is a dense subspace of X. Applying a recent degree theory of Kartsatos and Skrypnik, we establish the existence of an eigensolution to the nonlinear inclusion 0 ∈ T _x + λ C _x , with a regularization method by means of the duality operator. Moreover, possible branches of eigensolutions to the above inclusion are discussed. Furthermore, we give a surjectivity result about the operator λT + C when λ is not an eigenvalue for the pair (T, C), T being single-valued and densely defined.     

An inverse polynomial method for the identification of the leading coefficient in the Sturm–Liouville operator from boundary measurements

An inverse polynomial method of determining the unknown leading coefficient k=k(x) of the linear Sturm–Liouville operator Au=−(k(x)u^′(x))^′+q(x)u(x), x(0,1), is presented. As an additional condition only two measured data at the boundary (x=0,x=1) are used. In absence of a singular point (u^′(x)≠0,u^″(x)≠0,x[0,1]) the inverse problem is classified as a well-conditioned . If there exists at least one singular point, then the inverse problem is classified as moderately ill-conditioned (u^′(x₀)=0,x₀(0,1);u^′(x)≠0,x≠x₀;u^″(x)≠0,x[0,1]) and severely ill-conditioned (u^′(x₀)=u^″(x₀)=0,x₀(0,1);u^′(x)≠0,u^″(x)≠0,x≠x₀). For each of the cases direct problem solution is approximated by corresponding polynomials and the inverse problem is reformulated as a Cauchy problem for to the first order differential equation with respect the unknown function k=k(x). An approximate analytical solution of the each Cauchy problems are derived in explicit form. Numerical simulations all the above cases are given for noise free and noisy data. An accuracy of the presented approach is demonstrated on numerical test solutions.