The Signorini problem is an
elastostatics problem in
linear elasticity
Linear elasticity is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mec ...
: it consists in finding the
elastic equilibrium configuration of an
anisotropic non-homogeneous elastic body, resting on a
rigid friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction:
*Dry friction is a force that opposes the relative lateral motion of ...
less
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
and subject only to its
mass forces. The name was coined by
Gaetano Fichera
Gaetano Fichera (8 February 1922 – 1 June 1996) was an Italian mathematician, working in mathematical analysis, linear elasticity, partial differential equations and several complex variables. He was born in Acireale, and died in Rome.
Biog ...
to honour his teacher,
Antonio Signorini: the original name coined by him is problem with ambiguous
boundary conditions.
History
![Classical Signorini problem](https://upload.wikimedia.org/wikipedia/commons/b/b3/Classical_Signorini_problem.svg)
The problem was posed by
Antonio Signorini during a course taught at the ''
Istituto Nazionale di Alta Matematica'' in 1959, later published as the article , expanding a previous short exposition he gave in a note published in 1933. himself called it ''problem with ambiguous
boundary conditions'', since there are two alternative sets of
boundary conditions the solution ''must satisfy'' on any given
contact point. The statement of the problem involves not only
equalities ''but also
inequalities
Inequality may refer to:
Economics
* Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy
* Economic inequality, difference in economic well-being between population groups
* ...
'', and ''it is not
a priori
("from the earlier") and ("from the later") are Latin phrases used in philosophy to distinguish types of knowledge, justification, or argument by their reliance on empirical evidence or experience. knowledge is independent from current ...
known what of the two sets of boundary conditions is satisfied at each point''. Signorini asked to determine if the problem is
well-posed
The mathematical term well-posed problem stems from a definition given by 20th-century French mathematician Jacques Hadamard. He believed that mathematical models of physical phenomena should have the properties that:
# a solution exists,
# the sol ...
or not in a physical sense, i.e. if its solution exists and is unique or not: he explicitly invited young
analysts to study the problem.
Gaetano Fichera
Gaetano Fichera (8 February 1922 – 1 June 1996) was an Italian mathematician, working in mathematical analysis, linear elasticity, partial differential equations and several complex variables. He was born in Acireale, and died in Rome.
Biog ...
and
Mauro Picone attended the course, and Fichera started to investigate the problem: since he found no references to similar problems in the theory of
boundary value problems, he decided to approach it by starting from
first principle
In philosophy and science, a first principle is a basic proposition or assumption that cannot be deduced from any other proposition or assumption.
First principles in philosophy are from First Cause attitudes and taught by Aristotelians, and nua ...
s, specifically from the
virtual work principle.
During Fichera's researches on the problem, Signorini began to suffer serious health problems: nevertheless, he desired to know the answer to his question before his death. Picone, being tied by a strong friendship with Signorini, began to chase Fichera to find a solution: Fichera himself, being tied as well to Signorini by similar feelings, perceived the last months of 1962 as worrying days. Finally, on the first days of January 1963, Fichera was able to give a complete proof of the existence of a unique solution for the problem with ambiguous boundary condition, which he called the "Signorini problem" to honour his teacher. A preliminary research announcement, later published as , was written up and submitted to Signorini exactly a week before his death. Signorini expressed great satisfaction to see a solution to his question.
A few days later, Signorini had with his
family Doctor
Family medicine is a medical specialty within primary care that provides continuing and comprehensive health care for the individual and family across all ages, genders, diseases, and parts of the body. The specialist, who is usually a primary ...
, Damiano Aprile, the conversation quoted above.
The solution of the Signorini problem coincides with the birth of the field of
variational inequalities In mathematics, a variational inequality is an inequality involving a functional, which has to be solved for all possible values of a given variable, belonging usually to a convex set. The mathematical theory of variational inequalities was initi ...
.
Formal statement of the problem
The content of this section and the following subsections follows closely the treatment of
Gaetano Fichera
Gaetano Fichera (8 February 1922 – 1 June 1996) was an Italian mathematician, working in mathematical analysis, linear elasticity, partial differential equations and several complex variables. He was born in Acireale, and died in Rome.
Biog ...
in , and also : his derivation of the problem is different from
Signorini's one in that he does not consider only
incompressible bodies and a plane rest
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
, as Signorini does.
[See ) for the original approach.] The problem consist in finding the
displacement vector
In geometry and mechanics, a displacement is a vector whose length is the shortest distance from the initial to the final position of a point P undergoing motion. It quantifies both the distance and direction of the net or total motion along a ...
from the
natural configuration of an
anisotropic non-homogeneous elastic body that lies in a
subset of the three-
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
al
euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
whose
boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
* ''Boundaries'' (2016 film), a 2016 Canadian film
* ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film
*Boundary (cricket), the edge of the pla ...
is
and whose
interior normal is the
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
''
'', resting on a
rigid frictionless surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
whose
contact
Contact may refer to:
Interaction Physical interaction
* Contact (geology), a common geological feature
* Contact lens or contact, a lens placed on the eye
* Contact sport, a sport in which players make contact with other players or objects
* ...
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
(or more generally contact
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
) is
and subject only to its
body force
In physics, a body force is a force that acts throughout the volume of a body.
Springer site - Book 'Solid mechanics'preview paragraph 'Body forces'./ref>
Forces due to gravity, electric fields and magnetic fields are examples of body forces. Bo ...
s
, and
surface force
Surface force denoted ''fs'' is the force that acts across an internal or external surface element in a material body. Surface force can be decomposed into two perpendicular components: normal forces and shear forces. A normal force acts normal ...
s
applied on the free (i.e. not in contact with the rest surface) surface
: the set
and the contact surface
characterize the natural configuration of the body and are known a priori. Therefore, the body has to satisfy the general
equilibrium equations
:
written using the
Einstein notation as all in the following development, the ordinary
boundary conditions on
:
and the following two sets of
boundary conditions on
, where
is the
Cauchy stress tensor
In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that completely ...
. Obviously, the body forces and surface forces cannot be given in arbitrary way but they must satisfy a condition in order for the body to reach an equilibrium configuration: this condition will be deduced and analyzed in the following development.
The ambiguous boundary conditions
If ''
'' is any
tangent vector
In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are e ...
to the
contact
Contact may refer to:
Interaction Physical interaction
* Contact (geology), a common geological feature
* Contact lens or contact, a lens placed on the eye
* Contact sport, a sport in which players make contact with other players or objects
* ...
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
, then the ambiguous boundary condition in each
point
Point or points may refer to:
Places
* Point, Lewis, a peninsula in the Outer Hebrides, Scotland
* Point, Texas, a city in Rains County, Texas, United States
* Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland
* Point ...
of this set are expressed by the following two systems of
inequalities
Inequality may refer to:
Economics
* Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy
* Economic inequality, difference in economic well-being between population groups
* ...
:
or
Let's analyze their meaning:
*Each
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of conditions consists of three
relations,
equalities or
inequalities
Inequality may refer to:
Economics
* Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy
* Economic inequality, difference in economic well-being between population groups
* ...
, and all the second members are the
zero function
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
.
*The
quantities
Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a unit ...
at first member of each first relation are
proportional to the
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
of the
component
Circuit Component may refer to:
•Are devices that perform functions when they are connected in a circuit.
In engineering, science, and technology Generic systems
* System components, an entity with discrete structure, such as an assem ...
of the
displacement vector
In geometry and mechanics, a displacement is a vector whose length is the shortest distance from the initial to the final position of a point P undergoing motion. It quantifies both the distance and direction of the net or total motion along a ...
directed along the
normal vector .
*The quantities at first member of each second relation are proportional to the norm of the component of the
tension vector directed along the
normal vector ,
*The quantities at the first member of each third relation are proportional to the norm of the component of the tension vector along any
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
in the given
point
Point or points may refer to:
Places
* Point, Lewis, a peninsula in the Outer Hebrides, Scotland
* Point, Texas, a city in Rains County, Texas, United States
* Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland
* Point ...
to the
contact
Contact may refer to:
Interaction Physical interaction
* Contact (geology), a common geological feature
* Contact lens or contact, a lens placed on the eye
* Contact sport, a sport in which players make contact with other players or objects
* ...
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
.
*The quantities at the first member of each of the three relations are
positive
Positive is a property of positivity and may refer to:
Mathematics and science
* Positive formula, a logical formula not containing negation
* Positive number, a number that is greater than 0
* Plus sign, the sign "+" used to indicate a posi ...
if they have the same
sense
A sense is a biological system used by an organism for sensation, the process of gathering information about the world through the detection of stimuli. (For example, in the human body, the brain which is part of the central nervous system re ...
of the
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
they are
proportional to, while they are
negative if not, therefore the
constants of proportionality are respectively
and
.
Knowing these facts, the set of conditions applies to
point
Point or points may refer to:
Places
* Point, Lewis, a peninsula in the Outer Hebrides, Scotland
* Point, Texas, a city in Rains County, Texas, United States
* Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland
* Point ...
s of the
boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
* ''Boundaries'' (2016 film), a 2016 Canadian film
* ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film
*Boundary (cricket), the edge of the pla ...
of the body which ''do not'' leave the
contact
Contact may refer to:
Interaction Physical interaction
* Contact (geology), a common geological feature
* Contact lens or contact, a lens placed on the eye
* Contact sport, a sport in which players make contact with other players or objects
* ...
set
in the
equilibrium configuration, since, according to the first
relation, the
displacement vector
In geometry and mechanics, a displacement is a vector whose length is the shortest distance from the initial to the final position of a point P undergoing motion. It quantifies both the distance and direction of the net or total motion along a ...
''has no
component
Circuit Component may refer to:
•Are devices that perform functions when they are connected in a circuit.
In engineering, science, and technology Generic systems
* System components, an entity with discrete structure, such as an assem ...
s'' directed as the
normal vector , while, according to the second relation, the
tension vector ''may have a component'' directed as the normal vector
and having the same
sense
A sense is a biological system used by an organism for sensation, the process of gathering information about the world through the detection of stimuli. (For example, in the human body, the brain which is part of the central nervous system re ...
. In an analogous way, the set of conditions applies to points of the boundary of the body which ''leave'' that set in the equilibrium configuration, since displacement vector
''has a component'' directed as the normal vector
, while the
tension vector ''has no components'' directed as the normal vector
. For both sets of conditions, the tension vector has no tangent component to the
contact
Contact may refer to:
Interaction Physical interaction
* Contact (geology), a common geological feature
* Contact lens or contact, a lens placed on the eye
* Contact sport, a sport in which players make contact with other players or objects
* ...
set, according to the
hypothesis
A hypothesis (plural hypotheses) is a proposed explanation for a phenomenon. For a hypothesis to be a scientific hypothesis, the scientific method requires that one can test it. Scientists generally base scientific hypotheses on previous obse ...
that the body rests on a rigid ''frictionless'' surface.
Each system expresses a unilateral constraint, in the sense that they express the physical impossibility of the
elastic body to penetrate into the surface where it rests: the ambiguity is not only in the unknown values non-
zero
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
quantities must satisfy on the
contact
Contact may refer to:
Interaction Physical interaction
* Contact (geology), a common geological feature
* Contact lens or contact, a lens placed on the eye
* Contact sport, a sport in which players make contact with other players or objects
* ...
set but also in the fact that it is not a priori known if a point belonging to that set satisfies the system of boundary conditions or . The set of points where is satisfied is called the area of support of the elastic body on
, while its
complement respect to is called the area of separation.
The above formulation is ''general'' since the
Cauchy stress tensor
In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that completely ...
i.e. the
constitutive equation
In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and app ...
of the
elastic body has not been made explicit: it is equally valid assuming the
hypothesis
A hypothesis (plural hypotheses) is a proposed explanation for a phenomenon. For a hypothesis to be a scientific hypothesis, the scientific method requires that one can test it. Scientists generally base scientific hypotheses on previous obse ...
of
linear elasticity
Linear elasticity is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mec ...
or the ones of
nonlinear elasticity
In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal stra ...
. However, as it would be clear from the following developments, the problem is inherently
nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...
, therefore ''assuming a
linear stress tensor does not simplify the problem''.
The form of the stress tensor in the formulation of Signorini and Fichera
The form assumed by
Signorini and
Fichera for the
elastic potential energy
Elastic energy is the mechanical potential energy stored in the configuration of a material or physical system as it is subjected to elastic deformation by work performed upon it. Elastic energy occurs when objects are impermanently compressed, ...
is the following one (as in the previous developments, the
Einstein notation is adopted)
:
where
*
is the
elasticity tensor
In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of ...
*
is the
infinitesimal strain tensor
The
Cauchy stress tensor
In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that completely ...
has therefore the following form
:
and it is ''
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
'' with respect to the components of the infinitesimal strain tensor; however, it is not
homogeneous nor
isotropic.
Solution of the problem
As for the section on the formal statement of the Signorini problem, the contents of this section and the included subsections follow closely the treatment of
Gaetano Fichera
Gaetano Fichera (8 February 1922 – 1 June 1996) was an Italian mathematician, working in mathematical analysis, linear elasticity, partial differential equations and several complex variables. He was born in Acireale, and died in Rome.
Biog ...
in , , and also : obviously, the exposition focuses on the basics steps of the proof of the existence and uniqueness for the solution of problem , , , and , rather than the technical details.
The potential energy
The first step of the analysis of Fichera as well as the first step of the analysis of
Antonio Signorini in is the analysis of the potential energy, i.e. the following
functional
:
where
belongs to the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of admissible displacements
i.e. the set of
displacement vector
In geometry and mechanics, a displacement is a vector whose length is the shortest distance from the initial to the final position of a point P undergoing motion. It quantifies both the distance and direction of the net or total motion along a ...
s satisfying the system of
boundary conditions or . The meaning of each of the three terms is the following
*the first one is the total
elastic potential energy
Elastic energy is the mechanical potential energy stored in the configuration of a material or physical system as it is subjected to elastic deformation by work performed upon it. Elastic energy occurs when objects are impermanently compressed, ...
of the
elastic body
*the second one is the total
potential energy due to the
body force
In physics, a body force is a force that acts throughout the volume of a body.
Springer site - Book 'Solid mechanics'preview paragraph 'Body forces'./ref>
Forces due to gravity, electric fields and magnetic fields are examples of body forces. Bo ...
s, for example the
gravitational force
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the strong ...
*the third one is the potential energy due to
surface force
Surface force denoted ''fs'' is the force that acts across an internal or external surface element in a material body. Surface force can be decomposed into two perpendicular components: normal forces and shear forces. A normal force acts normal ...
s, for example the
forces exerted by the
atmospheric pressure
Atmospheric pressure, also known as barometric pressure (after the barometer), is the pressure within the atmosphere of Earth. The standard atmosphere (symbol: atm) is a unit of pressure defined as , which is equivalent to 1013.25 millibars, ...
was able to prove that the admissible displacement
which
minimize the integral
is a solution of the problem with ambiguous boundary conditions , , , and , provided it is a
function supported on the
closure of the set
: however
Gaetano Fichera
Gaetano Fichera (8 February 1922 – 1 June 1996) was an Italian mathematician, working in mathematical analysis, linear elasticity, partial differential equations and several complex variables. He was born in Acireale, and died in Rome.
Biog ...
gave a class of
counterexample
A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is a ...
s in showing that in general, admissible displacements are
not
smooth functions of these class. Therefore, Fichera tries to minimize the
functional in a wider
function space: in doing so, he first calculates the
first variation (or
functional derivative
In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional (a functional in this sense is a function that acts on functions) to a change in a function on ...
) of the given functional in the
neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographically localised community ...
of the sought minimizing admissible displacement
, and then requires it to be greater than or equal to
zero
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
:
Defining the following functionals
:
and
:
the preceding
inequality
Inequality may refer to:
Economics
* Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy
* Economic inequality, difference in economic well-being between population groups
* ...
is can be written as
:
This inequality is the
variational inequality for the Signorini problem.
See also
*
Linear elasticity
Linear elasticity is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mec ...
*
Variational inequality
Notes
References
Historical references
*.
*. A brief research survey describing the field of variational inequalities.
*. The encyclopedia entry about problems with unilateral constraints (the class of
boundary value problems the Signorini problem belongs to) he wrote for the ''Handbuch der Physik'' on invitation by
Clifford Truesdell.
*. ''The birth of the theory of variational inequalities remembered thirty years later'' (English translation of the contribution title) is an historical paper describing the beginning of the theory of variational inequalities from the point of view of its founder.
* . A volume collecting almost all works of Gaetano Fichera in the fields of
history of mathematics
The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments ...
and scientific divulgation.
* , (vol. 1), (vol. 2), (vol. 3). Three volumes collecting Gaetano Fichera's most important mathematical papers, with a biographical sketch of
Olga A. Oleinik.
* . A volume collecting Antonio Signorini's most important works with an introduction and a commentary of
Giuseppe Grioli.
Research works
*.
*. A short research note announcing and describing (without proofs) the solution of the Signorini problem.
*. The first paper where aa
existence
Existence is the ability of an entity to interact with reality. In philosophy, it refers to the ontological property of being.
Etymology
The term ''existence'' comes from Old French ''existence'', from Medieval Latin ''existentia/exsistentia' ...
and
uniqueness theorem In mathematics, a uniqueness theorem, also called a unicity theorem, is a theorem asserting the uniqueness of an object satisfying certain conditions, or the equivalence of all objects satisfying the said conditions. Examples of uniqueness theorems ...
for the Signorini problem is proved.
* . An English translation of the previous paper.
*.
*.
External links
*
Alessio Figalli, On global homogeneous solutions to the Signorini problem
{{DEFAULTSORT:Signorini Problem
Calculus of variations
Continuum mechanics
Elasticity (physics)
Partial differential equations