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The behavior of solutions of a second-order elliptic equation near a distinguished piece of the boundary is studied. On the remaining part of the boundary, the solutions are assumed to satisfy the homogeneous Dirichlet conditions. A necessary and sufficient condition is established for the existence of an L2 boundary value on the distinguished part of the boundary. Under the conditions of this criterion, estimates for the nontangential maximal function of the solution hold, the solution belongs to the space of (n − 1)-dimensionally continuous functions, and the boundary value is taken in a much stronger sense.

We consider a Steklov-type problem for the Laplace operator in a half-strip containing a small hole with the Dirichlet conditions on the lateral boundaries and the boundary of the hole and the Steklov spectral condition on the base of the half-strip. We prove that eigenvalues of this problem vanish as the small parameter (the “diameter” of the hole) tends to zero.

In this paper, a mass conserved domain decomposition method is developed for solving the nonlinear parabolic equations with Dirichlet conditions. The domain decomposition is divided into multiple subdomains, two steps are used to solve the numerical solutions on each sub-domain at every time interval [tn, tn+1]. In the first step, the interface fluxes are computed by local multi-points weighted average from the solutions. Secondly, the interior solutions are solved by the solution-flux coupled scheme. By some auxiliary lemmas and the Brouwer fixed point theorem, the existence, uniqueness and convergence of our scheme are analyzed in the discrete L2-norm. Numerical experiments are performed to illustrate the result.

Solving linear divergence-curl system with Dirichlet conditions is reduced to finding an unknown vector function in the space of piecewise-polynomial gradients of harmonic functions. In this approach one can use the boundary least squares method with a harmonic basis of a high order of approximation formulated by the authors previously. The justification of this method is given. The properties of the bilinear form and approximating properties of the basis are investigated. Convergence of approximate solutions is proved. A numerical example with estimates of experimental orders of convergence in $\begin{array}{} {\bf V}_h^p \end{array}$-norm for different parameters h, p (p ⩽ 10) is presented. The method does not require specification of penalty weight function.

We formulate a methodology to enforce interface conditions preserving higher-order continuity across the interface. Isogeometrical methods (IGA) naturally allow us to deal with equations of higher-order omitting the usage of mixed approaches. For multi-patch analysis of Kirchhoff–Love shell elements, G 1 continuity at the interface is required and serve here as a prototypical example for a higher-order coupling conditions. When working with this class of shell elements, two different types of constraints arise: Higher-order Dirichlet conditions and higher-order patch coupling conditions. A basis modification approach is presented here, based on a least-square formulation and the incorporation of the constraints into the IGA approximation space. An alternative formulation using Lagrange multipliers which are statically condensed via a discrete Null-Space method provides additional insight into the proposed formulation. A detailed comparison with a classical mortar approach shows the similarities and differences. Eventually, numerical examples demonstrate the capabilities of the presented formulation.

We consider a non-self-adjoint third-order operator on the interval [0,2] with real 1-periodic coefficients and three-point Dirichlet conditions at the points 0, 1 and 2. The eigenvalues of this op...

The paper deals with second order parabolic equations on bounded domains with Dirichlet conditions in arbitrary Euclidean spaces. Their interest comes from being models for describing reaction–diffusion processes in several frameworks. A linear diffusion term in divergence form is included which generates a strongly elliptic differential operator. A further linear part, of integral type, is present which accounts of nonlocal diffusion behaviours. The main result provides a unifying method for studying the existence and localization of solutions satisfying nonlocal associated boundary conditions. The Cauchy multipoint and the mean value conditions are included in this investigation. The problem is transformed into its abstract setting and the proofs are based on the homotopic invariance of the Leray–Schauder topological degree. A bounding function (i.e. Lyapunov-like function) theory is developed, which is new in this infinite dimensional context. It allows that the associated vector fields have no fixed points on the boundary of their domains and then it makes possible the use of a degree argument.

This paper is devoted to the functional analytic approach to the problem of existence of Markov processes in probability theory. More precisely, we construct Feller semigroups with Dirichlet conditions for second-order, uniformly elliptic integro-differential operators with discontinuous coefficients. Intuitively, we prove that there exists a Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle moves both by jumps and continuously in the state space until it dies at the time when it reaches the boundary. Our approach is based on real analysis techniques.

The present paper is devoted to the study of the existence of a solution u for a quasilinear second order differential equation with homogeneous Dirichlet conditions, where the right-hand side tends to infinity at u = 0 . The problem has been considered by several authors since the 70's. Mainly, nonnegative right-hand sides were considered and thus only nonnegative solutions were possible. Here we consider the case where the right-hand side can change sign but is non negative (finite or infinite) at u = 0 , while no restriction on its growth at u = 0 is assumed on its positive part. We show that there exists a nonnegative solution in a sense introduced in the paper; moreover, this solution is stable with respect to the right-hand side and is unique if the right-hand side is nonincreasing in u. We also show that if the right-hand side goes to infinity at zero faster than 1 / | u | , then only nonnegative solutions are possible. We finally prove by means of the study of a one-dimensional example that nonnegative solutions and even many solutions which change sign can exist if the growth of the right-hand side is 1 / | u | γ with 0 γ 1 .

Made available in DSpace on 2021-07-14T10:31:21Z (GMT). No. of bitstreams: 0 Previous issue date: 2021-04-16. Added 1 bitstream(s) on 2021-07-14T11:32:06Z : No. of bitstreams: 1 S1806-11172021000100101.pdf: 593269 bytes, checksum: 1180692aa601e48a7902c3868089a22b (MD5) It is presented a way to obtain the closed form for Green’s function related to the nonhomogeneous one-dimensional Helmholtz equation with homogeneous Dirichlet conditions on the boundary of the domain from its Fourier sine series representation. A closed form for the sum of the series ∑k=1∞sin⁡k⁢x⁢sin⁡k⁢y/(k2-α2) is found in the process. Universidade Estadual Paulista “Júlio de Mesquita Filho”, Departamento de Física Universidade Estadual Paulista “Júlio de Mesquita Filho”, Departamento de Física

In the present article, we consider a porous thermoelastic system with distributed delay term in second sound, by using the energy method combined with multiplicative technique, we show the polynomial decay estimate of (1) with (3) in Theorem 2.7 and exponential stability of (1) with conditions (38) in Theorem 3.4. A new restriction on delay term depending on the time in (4) is used to show that the solution energy should be stable. The importance of the present research lies with describing the role of Dirichlet conditions in the nature of stability.

We study a generalization of the power of Laplace operator with null Dirichlet conditions by means of the spectral theory and prove several existence results for nonlinear equations with such operators, especially when the problem is resonant. Some regularity results are also obtained.

We give a rather short and self-contained presentation of the global existence for Leray-Hopf weak solutions to the three dimensional incompressible Navier-Stokes equations, with constant density. We give a unified treatment in terms of the domains and the relative boundary conditions and in terms of the approximation methods. More precisely, we consider the case of the whole space, the flat torus, and the case of a general bounded domain with a smooth boundary (the latter supplemented with homogeneous Dirichlet conditions). We consider as approximation schemes the Leray approximation method, the Faedo-Galerkin method, the semi-discretization in time and the approximation by adding a Smagorinsky-Ladyžhenskaya term. We mainly focus on developing a unified treatment especially in the compactness argument needed to show that approximations converge to the weak solutions.

In this study, we consider the Fisher equation in bounded domains. By Faedo–Galerkin’s method and with a homogeneous Dirichlet conditions, the existence of a global solution is proved.

The study concerns the winding head thermal design of electrical machines in difficult thermal environments. The new approach is adapted for all basic shapes and solves the thermal behaviour of a random wire layout. The model uses the nodal method but does not use the common homogenization method for the winding slot. The layout impact can be precisely studied to find different hotspots. To achieve this a Delaunay triangulation provides the thermal links between adjoining wires in the slot. Vorono&iuml, tessellation gives a cutting to estimate thermal conductance between adjoining wires. This thermal behaviour is simulated in cell cutting and it is simplified with the thermal bridge notion to obtain a simple solving of these thermal conductances. The boundaries are imposed on the slot borders with Dirichlet condition. Then solving with many Dirichlet conditions is described. Some results show different possible applications with rectangular and round shapes, one ore many boundaries, different limit condition values and different layouts. The model can be integrated into a larger model that represents the stator to have best results.

The scalar wave scattering by the circular slot in the infinitely thin screen is considered in the case of the Dirichlet boundary conditions. Here, we represent the circular slot as two separate scattering objects, which interact with each other. One of them is a circular disc. Another one is a screen with a circular hole. The operator equations are obtained relatively amplitudes of the scattered field. The dependence of the total scattering cross section on the frequency and diffraction patterns are represented.

Consider the diffusive Hamilton–Jacobi equation $$\begin{aligned} u_t-\Delta u=|\nabla u|^p+h(x)\ \ \text { in } \Omega \times (0,T) \end{aligned}$$ with Dirichlet conditions, which arises in stochastic control problems as well as in KPZ type models. We study the question of the gradient blowup rate for classical solutions with $$p>2$$ . We first consider the case of time-increasing solutions. For such solutions, the precise rate was obtained by Guo and Hu (2008) in one space dimension, but the higher dimensional case has remained an open question (except for radially symmetric solutions in a ball). Here, we partially answer this question by establishing the optimal estimate 1 $$\begin{aligned} C_1(T-t)^{-1/(p-2)}\le \Vert \nabla u(t)\Vert _\infty \le C_2(T-t)^{-1/(p-2)} \end{aligned}$$ for time-increasing gradient blowup solutions in any convex, smooth bounded domain $$\Omega $$ with $$22$$ , we show that more singular rates may occur for solutions which are not time-increasing. Namely, for a suitable class of solutions in one space-dimension, we prove the lower estimate $$\Vert u_x(t)\Vert _\infty \ge C(T-t)^{-2/(p-2)}$$ .

We consider the dynamics of a quantum particle of mass $m$ on a $n$-edges star-graph with Hamiltonian $H_K=-(2m)^{-1}\hbar^2 \Delta$ and Kirchhoff conditions in the vertex. We describe the semiclassical limit of the quantum evolution of an initial state supported on one of the edges and close to a Gaussian coherent state. We define the limiting classical dynamics through a Liouville operator on the graph, obtained by means of Kre\u{\i}n's theory of singular perturbations of self-adjoint operators. For the same class of initial states, we study the semiclassical limit of the wave and scattering operators for the couple $(H_K,H_{D}^{\oplus})$, where $H_{D}^{\oplus}$ is the free Hamiltonian with Dirichlet conditions in the vertex., Comment: 31 pages

This paper introduces a new approach to tackle the boundary value problem representing magnetohydrodynamics flow developed by a rotating disk. The problem comprises of a coupled and non-linear differential system which appears difficult to be computed exactly. A contemporary MATLAB’s package bvp4c is used to solve the problem numerically and then, with the help of the numerical solution, a closed-form solution is constructed via a systematic method. All computations are made for a specific value of the magnetic interaction parameter. Residuals of governing equations are plotted which ascertain that the obtained closed-form solution matches closely with the exact solution.

This paper deals with positive solutions of the fully parabolic system { u t = Δ u − χ ∇ ⋅ ( u ∇ v ) in Ω × ( 0 , ∞ ) , τ 1 v t = Δ v − v + w in Ω × ( 0 , ∞ ) , τ 2 w t = Δ w − w + u in Ω × ( 0 , ∞ ) under mixed boundary conditions (no-flux and Dirichlet conditions) in a smooth bounded convex domain Ω ⊂ R 4 with positive parameters τ 1 , τ 2 , χ > 0 and nonnegative smooth initial data ( u 0 , v 0 , w 0 ) . Global existence and boundedness of solutions were shown if ‖ u 0 ‖ L 1 ( Ω ) ( 8 π ) 2 / χ in Fujie–Senba (2017). In the present paper, it is shown that there exist blowup solutions satisfying ‖ u 0 ‖ L 1 ( Ω ) > ( 8 π ) 2 / χ . This result suggests that the system can be regard as a generalization of the Keller–Segel system, which has 8 π / χ -dichotomy. The key ingredients are a Lyapunov functional and quantization properties of stationary solutions of the system in R 4 .

After investigating existence and uniqueness of the global strong solutions for Euler-Voigt equations under Dirichlet conditions, we obtain the Kato's type theorems for the convergence of the Euler-Voigt equations to Euler equations. More precisely, the necessary and sufficient conditions that the solution of Euler-Voigt equation converges to the one of Euler equations, as \begin{document}$ \alpha\to 0 $\end{document} , can be obtained.

This paper is devoted to the study of a nonlinear Kirchhoff-Carrier wave equation in an annular associated with nonhomogeneous Dirichlet conditions. At first, by applying the Faedo-Galerkin, we prove existence and uniqueness of the solution of the problem considered. Next, by constructing Lyapunov functional, we prove a blow-up result for solutions with a negative initial energy and establish a sufficient condition to obtain the exponential decay of weak solutions.

We are concerned with Holder regularity estimates for weak solutions \begin{document}$u$\end{document} to nonlocal Schrodinger equations subject to exterior Dirichlet conditions in an open set \begin{document}$\Omega\subset \mathbb{R}^N$\end{document} . The class of nonlocal operators considered here are defined, via Dirichlet forms, by symmetric kernels \begin{document}$K(x, y)$\end{document} bounded from above and below by \begin{document}$|x-y|^{-N-2s}$\end{document} , with \begin{document}$s\in (0, 1)$\end{document} . The entries in the equations are in some Morrey spaces and the domain \begin{document}$\Omega$\end{document} satisfies some mild regularity assumptions. In the particular case of the fractional Laplacian, our results are new. When \begin{document}$K$\end{document} defines a nonlocal operator with sufficiently regular coefficients, we obtain Holder estimates, up to the boundary of \begin{document}$ \Omega$\end{document} , for \begin{document}$u$\end{document} and the ratio \begin{document}$u/d^s$\end{document} , with \begin{document}$d(x) = \text{dist}(x, \mathbb{R}^N\setminus\Omega)$\end{document} . If the kernel \begin{document}$K$\end{document} defines a nonlocal operator with Holder continuous coefficients and the entries are Holder continuous, we obtain interior \begin{document}$C^{2s+\beta}$\end{document} regularity estimates of the weak solutions \begin{document}$u$\end{document} . Our argument is based on blow-up analysis and compact Sobolev embedding.

In this work we analyze the behavior of the solutions to nonlocal evolution equations of the form u t ( x , t ) = ∫ J ( x − y ) u ( y , t ) d y − h e ( x ) u ( x , t ) + f ( x , u ( x , t ) ) with x in a perturbed domain Ω e ⊂ Ω which is thought as a fixed set Ω from where we remove a subset A e called the holes. We choose appropriated families of functions h e ∈ L ∞ in order to deal with both Neumann and Dirichlet conditions in the holes setting a Dirichlet condition outside Ω. Moreover, we take J as a non-singular kernel and f as a nonlocal nonlinearity. Under the assumption that the characteristic functions of Ω e have a weak limit, we study the limit of the solutions providing a nonlocal homogenized equation.

We apply the recently developed least squares stabilized symmetric Nitsche method for enforcement of Dirichlet boundary conditions to the finite cell method. The least squares stabilized Nitsche method in combination with finite cell stabilization leads to a symmetric positive definite stiffness matrix and relies only on elementwise stabilization, which does not lead to additional fill in. We prove a priori error estimates and bounds on the condition numbers.

For a singularly perturbed parabolic problem with Dirichlet conditions we prove the existence of a solution periodic in time and with boundary layers at both ends of the space interval in the case that the degenerate equation has a double root. We construct the corresponding asymptotic expansion in the small parameter. It turns out that the algorithm of the construction of the boundary layer functions and the behavior of the solution in the boundary layers essentially differ from that ones in case of a simple root. We also investigate the stability of this solution and the corresponding region of attraction.

A linear wave equation with acoustic boundary conditions (ABC) on a portion of the boundary and Dirichlet conditions on the rest of the boundary is considered. The ABC contain a memory damping with respect to the normal displacement of the boundary point. In this paper, we establish polynomial energy decay rates for the wave equation by using resolvent estimates.

In this paper, an optimal boundary control problem for a distributed elliptic system on Lipschitz domains with boundary homogeneous Dirichlet conditions and independently with Neumann conditions is analysed. The necessary and sufficient optimality conditions for such problems with the quadratic cost functionals are obtained. A Jacobi spectral Galerkin method is introduced to develop a direct solution technique for the numerical solution of elliptic problems subject to Dirichlet and Neumann conditions in one and two dimensions. The numerical examples are included to demonstrate the validity and applicability of the techniques and comparison is made with the existing results. The method is easy to implement and yields very accurate results.

The Laplacian Eigenfunction method for fluid simulation, which we refer to as Eigenfluids , introduced an elegant new way to capture intricate fluid flows with near-zero viscosity. However, the approach does not scale well, as the memory cost grows prohibitively with the number of eigenfunctions. The method also lacks generality, because the dynamics are constrained to a closed box with Dirichlet boundaries, while open, Neumann boundaries are also needed in most practical scenarios. To address these limitations, we present a set of analytic eigenfunctions that supports uniform Neumann and Dirichlet conditions along each domain boundary, and show that by carefully applying the discrete sine and cosine transforms, the storage costs of the eigenfunctions can be made completely negligible. The resulting algorithm is both faster and more memory-efficient than previous approaches, and able to achieve lower viscosities than similar pseudo-spectral methods. We are able to surpass the scalability of the original Laplacian Eigenfunction approach by over two orders of magnitude when simulating rectangular domains. Finally, we show that the formulation allows forward scattering to be directed in a way that is not possible with any other method.

Two two-dimensional plates with bending described by Sophie Germain’s equation with the biharmonic operator are joined in the form of a cross with clamped ends, but with free lateral sides outside the cross core. Asymptotics of the deflection of the junction with respect to the relative width of the plates regarded as a small parameter is constructed and justified. The variational-asymptotic model includes a system of two ordinary differential equations of the fourth and second orders with Dirichlet conditions at the endpoints of the one-dimensional cross and the Kirchhoff transmission conditions at its center. They are derived by analyzing the boundary layer near the crossing of the plates and mean that the deflection and the angles of rotation at the central point are continuous and that the total bending force and the principal torques vanish. Possible generalizations of the asymptotic analysis are discussed.

In this paper, a maximum principle for the one-dimensional sub-diffusion equation with Atangana–Baleanu fractional derivative is formulated and proved. The proof of the maximum principle is based on an extremum principle for the Atangana–Baleanu fractional derivative that is given in the paper, too. The maximum principle is then applied to show that the initial–boundary-value problem for the linear and nonlinear time-fractional diffusion equations possesses at most one classical solution and this solution continuously depends on the initial and boundary conditions.

The local projection stabilization (LPS) method is already an established method for stabilizing saddle-point problems and convection–diffusion problems. The a priori error analysis is usually done for homogeneous Dirichlet data. It turns out that without Dirichlet conditions the situation is more involved, because additional boundary terms appear in the analysis. The standard approach can be modified by using additional stabilization terms on the non-Dirichlet boundary parts. We show that such terms lead to a similar a priori estimate as the classical LPS method but in a stronger norm.

A class of unsteady analytic solutions for the evolution of viscous Newtonian fluids over a superhydrophobic surface is derived. The surface is represented by regular rectilinear riblets and the fluid is assumed to flow along the grooves' direction. A mixed boundary condition is imposed on the surface, consisting in homogeneous Neumann conditions over the riblet voids and homogeneous Dirichlet conditions on the wall intervals. The transition between the above conditions is modelled through a Robin condition with non-constant smooth coefficients in a completely general manner. A global solution is derived and relevant examples, that can be fruitfully adopted as benchmark solutions for testing numerical solvers, are discussed. (C) 2017 Elsevier Inc. All rights reserved.

Let be a continuous function which increases to on an infinite interval of the form . A necessary and sufficient condition is found on a sequence increasing to which ensures that for each Dirichlet series of the form , , which is absolutely convergent in the following relation holds: where and are the maximum modulus and maximum term of the series, respectively. Bibliography: 10 titles.

This article is devoted to deduce the expression and the main properties of the Green’s function related to a general constant coefficients fractional difference equation coupled to Dirichlet conditions. In particular, it is proved that such function has constant sign on their set of definition and, moreover, it satisfies some additional strong sign conditions that are fundamental to define suitable cones, where to ensure the existence of solutions of nonlinear problems.

In this paper the finite time stabilization of an infinite network of vibrating strings is studied. We consider a network with N ∈ N ⁎ , N ≥ 2 finite edges and one infinite edge. This network is subject to homogeneous Dirichlet conditions at the endpoints of the finite edges. At the internal vertex 0, we put the continuity condition and a balanced damping condition of the form ∑ j = 0 N ( ∂ x u j ( 0 , t ) − ∂ t u j ( 0 , t ) ) = 0 . By a spectral analysis technique, we prove that when the finite edges lengths are rationally proportional, the energy of the system becomes constant within a finite time T e . We also prove that for an equilateral network we have T e = 2 l , where l is the common edge length. This is a finite time stabilization result of the system to a state with constant energy. This finite time limit energy is given in terms of the initial data and computed for a generic network. The main idea is to calculate the resolvent and to prove that it is of finite exponential type on some subspace of the energy space.

We study the wave equation on an unbounded network of \begin{document} $N, N∈\mathbb{N}^*$ \end{document} , finite strings and a semi-infinite one with a single vertex identified to 0. We consider continuity and dissipation conditions at the vertex and Dirichlet conditions at the extremities of the finite edges. The dissipation is given by a damping constant $α>0$ via the condition \begin{document} $\sum_{j = 0}^N\partial_xu_j(0, t) = α \partial_tu_0(0, t)$ \end{document} . We give a complete spectral description and we use it to study the energy decay of the solution. We prove that for \begin{document} $α\not = N+1$ \end{document} we have an exponential decay of the energy and we give an explicit formula for the decay rate when the finite edges have the same length.

In recent papers (see e.g. Ruas (2020a) and Ruas (2020b)) a nonparametric technique of the Petrov–Galerkin type was analyzed, whose aim is the accuracy enhancement of higher order finite element methods to solve boundary value problems with Dirichlet conditions, posed in smooth curved domains. In contrast to parametric elements, it employs straight-edged triangular or tetrahedral meshes fitting the domain. In order to attain best-possible orders greater than one, piecewise polynomial trial-functions are employed, which interpolate the Dirichlet conditions at points of the true boundary. The test-functions in turn are defined upon the standard degrees of freedom associated with the underlying method for polytopic domains. As a consequence, when the problem at hand is self-adjoint a non symmetric linear system has to be solved. This paper is primarily aimed at showing that in this case, an efficient symmetrization of the solution procedure can be achieved by means of a fast converging iterative method. In order to illustrate the great generality of our nonparametric approach, experimentation is presented with a finite element method having degrees of freedom other than nodal values. More specifically we consider a nonconforming quadratic element in the solution of the three-dimensional Poisson equation. The performance evaluation however is conducted as well for two versions of the classical conforming quadratic method, namely, the nonparametric Petrov–Galerkin formulation considered in Ruas (2020b) and the standard isoparametric one. The study of this symmetrization is completed by an optimal error estimation in the broken H 1 -norm for the nonparametric version of the nonconforming method, which had not been addressed in previous work.

The Friedland–Hayman inequality is a sharp inequality concerning the growth rates of homogeneous, harmonic functions with Dirichlet boundary conditions on complementary cones dividing Euclidean space into two parts. In this paper, we prove an analogous inequality in which one divides a convex cone into two parts, placing Neumann conditions on the boundary of the convex cone and Dirichlet conditions on the interface. This analogous inequality was already proved by us jointly with Sarah Raynor. Here, we present a new proof that permits us to characterize the case of equality. In keeping with the two-phase free boundary theory introduced by Alt, Caffarelli, and Friedman, such an improvement can be expected to yield further regularity in free boundary problems.

Bounds on the spectrum of the Schur complements of subdomain stiffness matrices with respect to the interior variables are key ingredients in the analysis of many domain decomposition methods. Here we are interested in the analysis of floating clusters, i.e. subdomains without prescribed Dirichlet conditions that are decomposed into still smaller subdomains glued on primal level in some nodes and/or by some averages. We give the estimates of the regular condition number of the Schur complements of the clusters arising in the discretization of problems governed by 2D Laplacian. The estimates depend on the decomposition and discretization parameters and gluing conditions. We also show how to plug the results into the analysis of H-TFETI methods and compare the estimates with numerical experiments. The results are useful for the analysis and implementation of powerful massively parallel scalable algorithms for the solution of variational inequalities.

We study the Dirichlet problem of the n-dimensional complex Monge—Ampere equation det(u ij ) = F/|z|2α, where 0 < α < n. This equation comes from La Nave—Tian’s continuity approach to the Analytic Minimal Model Program.

There exist many explicit evaluations of Dirichlet series. Most of them are constructed via the same approach: by taking products or powers of Dirichlet series with a known Euler product representation. In this paper we derive a result of a new flavour: we give the Dirichlet series representation to solution $f=f(s,w)$ of the functional equation $L(s-wf)=\exp(f)$, where $L(s)$ is the L-function corresponding to a completely multiplicative function. Our result seems to be a Dirichlet series analogue of the well known Lagrange-B\"urmann formula for power series. The proof is probabilistic in nature and is based on Kendall's identity, which arises in the fluctuation theory of L\'evy processes., Comment: 12 pages, 1 figure