Elliptic boundary value problem
In mathematics, an elliptic boundary value problem is a special kind of boundary value problem which can be thought of as the stable state of an evolution problem. For example, the Dirichlet problem for the Laplacian gives the eventual distribution of heat in a room several hours after the heating is turned on. Differential equations describe a large class of natural phenomena, from the heat equation describing the evolution of heat in (for instance) a metal plate, to the Navier-Stokes equation describing the movement of fluids, including Einstein's equations describing the physical universe in  a relativistic way. Although all these equations are boundary value problems, they are further subdivided into categories. This is necessary because each category must be analyzed using different techniques. The present article deals with the category of boundary value problems known as linear elliptic problems. Boundary value problems and partial differential equations specify relations between two or more quantities. For instance, in the heat equation, the rate of change of temperature at a point is related to the difference of temperature between that point and the nearby points so that, over time, the heat flows from hotter points to cooler  points. Boundary value problems can involve space, time and other quantities such as temperature, velocity, pressure, magnetic field, etc... Some problems do not involve time. For instance, if one hangs a clothesline between the house and a tree, then in the absence of wind, the clothesline will not move and will adopt a gentle hanging curved shape known as the catenary. This curved shape can be computed as the solution of a differential equation relating position, tension, angle and gravity, but since the shape does not change over time, there is no time variable. Elliptic boundary value problems are a  class of problems which do not involve the time variable, and instead only depend on space variables. The main example. In two dimensions, let formula_1 be the coordinates. We will use the notation formula_2 for the first and second partial derivatives of formula_3 with respect to formula_4, and a similar notation for formula_5. We will use the symbols formula_6 and formula_7 for the partial differential operators in formula_4 and formula_5. The second partial derivatives will be denoted formula_10 and formula_11. We also define the gradient formula_12, the Laplace operator formula_13 and the divergence formula_14. Note from the definitions that formula_15 . The main example for boundary value problems is the Laplace operator, where formula_18 is a region in the plane and formula_19 is the boundary of that region. The function formula_20 is known data and the solution formula_3 is what must be computed. This example has the same essential properties as all other elliptic boundary value problems. The solution formula_3 can be interpreted as the stationary or limit distribution of heat in a metal plate shaped like formula_18, if this metal plate has its boundary adjacent to ice (which is kept at zero degrees, thus the Dirichlet boundary condition.) The function  formula_20 represents the intensity of heat generation at each point in the plate (perhaps there is an electric heater resting on the metal plate, pumping heat into the plate at rate formula_25, which does not vary over time, but may be nonuniform in space on the metal plate.) After waiting for a long time, the temperature distribution in the metal plate will approach formula_3. Nomenclature. Let formula_27 where formula_28 and formula_29 are constants. formula_30 is called a second order differential operator. If we formally replace the derivatives formula_6 by formula_4 and formula_7 by formula_5, we obtain the expression If we  set this expression equal to some constant formula_36, then we obtain either an ellipse (if formula_37 are all the same sign) or a hyperbola (if formula_28 and formula_29 are of opposite signs.) For that reason, formula_40 is said to be elliptic when formula_41 and hyperbolic if formula_42. Similarly, the operator formula_43 leads to a parabola, and so this formula_40 is said to be parabolic. We now generalize the notion of ellipticity. While it may not be obvious that our generalization is the right one, it turns out that it does preserve most of the necessary properties for the purpose of  analysis. General linear elliptic boundary value problems of the second degree. Let formula_45 be the space variables. Let formula_46 be real valued functions of formula_47. Let formula_40 be a second degree linear operator. That is, We have used the subscript formula_51 to denote the partial derivative with respect to the space variable formula_52. The two formulae are equivalent, provided that In matrix notation, we can let formula_54 be an formula_55 matrix valued function of formula_4 and formula_57 be a formula_58-dimensional column vector-valued function of formula_4, and then we may write One may assume, without loss of generality, that  the matrix formula_28 is symmetric (that is, for all formula_62, formula_63. We make that assumption in the rest of this article. We say that the operator formula_40 is "elliptic" if, for some constant formula_65, any of the following equivalent conditions hold: An elliptic boundary value problem is then a system of equations like This particular example is the Dirichlet problem. The Neumann problem is where formula_73 is the derivative of formula_3 in the direction of the outwards pointing normal of formula_19. In general, if formula_76 is any trace operator, one can construct the boundary value problem In the rest of  this article, we assume that formula_40 is elliptic and that the boundary condition is the Dirichlet condition formula_80. Sobolev spaces. The analysis of elliptic boundary value problems requires some fairly sophisticated tools of functional analysis. We require the space formula_81, the Sobolev space of "once-differentiable" functions on formula_18, such that both the function formula_3 and its partial derivatives formula_84, formula_85 are all square integrable. There is a subtlety here in that the partial derivatives must be defined "in the weak sense" (see the article on Sobolev spaces for details.) The space formula_86 is a Hilbert space, which accounts for  much of the ease with which these problems are analyzed. The discussion in details of Sobolev spaces is beyond the scope of this article, but we will quote required results as they arise. Unless otherwise noted, all derivatives in this article are to be interpreted in the weak, Sobolev sense. We use the term "strong derivative" to refer to the classical derivative of calculus. We also specify that the spaces formula_87, formula_88 consist of functions that are formula_36 times strongly differentiable, and that the formula_36th derivative is continuous. Weak or variational formulation. The first step to cast the boundary value  problem as in the language of Sobolev spaces is to rephrase it in its weak form. Consider the Laplace problem formula_91. Multiply each side of the equation by a "test function" formula_92 and integrate by parts using Green's theorem to obtain We will be solving the Dirichlet problem, so that formula_94. For technical reasons, it is useful to assume that formula_92 is taken from the same space of functions as formula_3 is so we also assume that formula_97. This gets rid of the formula_98 term, yielding where If formula_40 is a general elliptic operator, the same reasoning leads to  the bilinear form We do not discuss the Neumann problem but note that it is analyzed in a similar way. Continuous and coercive bilinear forms. The map formula_104 is defined on the Sobolev space formula_105 of functions which are once differentiable and zero on the boundary formula_19, provided we impose some conditions on formula_107 and formula_18. There are many possible choices, but for the purpose of this article, we will assume that The reader may verify that the map formula_104 is furthermore bilinear and continuous, and that the map formula_119 is linear in formula_92, and continuous if (for instance) formula_20  is square integrable. We say that the map formula_122 is coercive if there is an formula_65 for all formula_124, This is trivially true for the Laplacian (with formula_126) and is also true for an elliptic operator if we assume formula_127 and formula_128. (Recall that formula_129 when formula_40 is elliptic.) Existence and uniqueness of the weak solution. One may show, via the Lax–Milgram lemma, that whenever formula_104 is coercive and formula_119 is continuous, then there exists a unique solution formula_133 to the weak problem (*). If further formula_104 is symmetric (i.e., formula_135), one can show the same result using the Riesz  representation theorem instead. This relies on the fact that formula_104 forms an inner product on formula_137, which itself depends on Poincaré's inequality. Strong solutions. We have shown that there is a formula_133 which solves the weak system, but we do not know if this formula_3 solves the strong system Even more vexing is that we are not even sure that formula_3 is twice differentiable, rendering the expressions formula_143 in formula_144 apparently meaningless. There are many ways to remedy the situation, the main one being regularity. Regularity. A regularity theorem for a linear elliptic boundary value problem of the second  order takes the form Theorem "If (some condition), then the solution formula_3 is in formula_146, the space of "twice differentiable" functions whose second derivatives are square integrable." There is no known simple condition necessary and sufficient for the conclusion of the theorem to hold, but the following conditions are known to be sufficient: It may be tempting to infer that if formula_19 is piecewise formula_148 then formula_3 is indeed in formula_153, but that is unfortunately false. Almost everywhere solutions. In the case that formula_154 then the second derivatives of formula_3 are defined almost everywhere, and in that case formula_156 almost  everywhere. Strong solutions. One may further prove that if the boundary of formula_157 is a smooth manifold and formula_20 is infinitely differentiable in the strong sense, then formula_3 is also infinitely differentiable in the strong sense. In this case, formula_156 with the strong definition of the derivative. The proof of this relies upon an improved regularity theorem that says that if formula_19 is formula_87 and formula_163, formula_164, then formula_165, together with a Sobolev imbedding theorem saying that functions in formula_166 are also in formula_167 whenever formula_168. Numerical solutions. While in exceptional circumstances, it is possible to solve elliptic problems explicitly , in general it is an impossible task. The natural solution is to approximate the elliptic problem with a simpler one and to solve this simpler problem on a computer. Because of the good properties we have enumerated (as well as many we have not), there are extremely efficient numerical solvers for linear elliptic boundary value problems (see finite element method, finite difference method and spectral method for examples.) Eigenvalues and eigensolutions. Another Sobolev imbedding theorem states that the inclusion formula_169 is a compact linear map. Equipped with the spectral theorem for compact linear operators, one obtains the following result. Theorem  Assume that formula_104 is coercive, continuous and symmetric. The map formula_171 from formula_172 to formula_172 is a compact linear map. It has a basis of eigenvectors formula_174 and matching eigenvalues formula_175 such that Series solutions and the importance of eigensolutions. If one has computed the eigenvalues and eigenvectors, then one may find the "explicit" solution of formula_156, via the formula where The series converges in formula_188. Implemented on a computer using numerical approximations, this is known as the spectral method. An example. Consider the problem The reader may verify that the eigenvectors are exactly with eigenvalues The Fourier coefficients of  formula_195 can be looked up in a table, getting formula_196. Therefore, yielding the solution Maximum principle. There are many variants of the maximum principle. We give a simple one. Theorem. "(Weak maximum principle.) Let formula_199, and assume that formula_200. Say that formula_201 in formula_18. Then formula_203. In other words, the maximum is attained on the boundary." A strong maximum principle would conclude that formula_204 for all formula_205 unless formula_3 is constant.